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ELEMENTARY  ALGEBRA 


BY 


WALTER   R.    MARSH 

HEAD  MASTER   PINGRY   SCHOOL,   ELIZABETH,   X.J. 


Of    TH€ 

UNIVERSITY 

or 


NEW   YOEK 

CHARLES   SCRIBNER'S   SONS 

1907 


M3f 


MAY  29  191 
GIFT 


COPYRIGHT,   1905,   1907,  BY 
CHARLES   SCRIBNER's   SONS 


PREFACE 

The  subject-matter  of  this  text  follows  the  require- 
ments of  the  College  Entrance  Examination  Board  both 
as  to  subjects  treated  as  well  as  to  those  omitted,  but 
especial  emphasis  is  placed  upon  those  principles  which 
are  the  tools  of  more  advanced  work  in  mathematics. 
The  philosophy  per  se  of  algebra  and  all  algebraic  puz- 
zles are  therefore  omitted,  to  give  place  to  a  logical  dis- 
cussion, simply  told,  of  the  fundamental  principles.  The 
scheme  of  the  whole  text  is  to  illustrate  the  meaning  of 
a  principle  by  carefully  selected  exercises;  every  prin- 
ciple is  followed  by  such  a  group  of  examples  as  will 
exact  a  mastery  of  the  principle  involved  before  another 
topic  is  taken  up.  The  examples  are  expressly  prepared 
to  illustrate  various  principles  treated  in  the  text.  Nearly 
a  thousand  of  these  examples  are  taken  from  the  most 
recent  college  entrance  papers. 

The  attention  of  teachers  is  especially  invited  to  the 
use  of  Graphical  Methods  throughout  the  book,  the  in- 
troduction of  the  Negative  Number,  the  treatment  of  the 
Graphs  of  Equations,  the  introduction  of  Equations  used 
in  Physics,  and  the  insertion  of  problems  from  Physics 
in  Ratio  and  in  Variation,  and  to  the  treatment  of  the 
Progressions  and  of  Permutations  and  Combinations. 

It  is  suggested  that  paragraphs,  exercises,  and  exam- 
ples marked  by  the  *  be  omitted  dt  first  reading. 

V 

219153 


VI  PREFACE 

The  author  begs  to  acknowledge  gratefully  the  valuable 
assistance  of  Professor  Charles  H.  Ashton  of  the  Univer- 
sity of  Kansas,  of  Miss  Mary  M.  Wardwell  of  the  Central 
High  School,  Buffalo,  N.Y.,  and  of  Mr.  Frank  C.  Rob- 
ertson of  the  Pingry  School,  Elizabeth,  N.  J.,  not  only  for 
their  careful  reading  of  the  proofs,  but  also  for  their 
criticisms  of  the  text. 


CONTENTS 

CHAPTER  PAGE 

I.  Introduction  and  Definitions 1 

II.  Addition  and  Subtraction 19 

III.  Multiplication  and  Division 30 

IV.  Equations  and  Problems 47 

V.  Type  Forms  in  Multiplication 65 

VI.  Factoring 75 

VII.  Highest  Common  Factors.     Lowest  Common  Multiples  100 

VIII.  Fractions 115 

IX.  Simple  Equations   ...,.•..  142 

X.  Graphs 158 

XI.  Simultaneous  Simple  Equations 163 

XII.  Problems  involving  Simple  Equations  .         .         •         .  188 

XIII.  Inequalities .203 

XIV.  Involution  and  Evolution 210 

XV.  Radicals          . 22G 

XVI.  Imaginaries 249 

XVII.  Theory  of  Exponents 254 

XVIII.  Quadratic  Equations 268 

XIX.  Simultaneous  Equations  solvable  by  Quadratics  .         .  299 

XX.  Problems  involving  Quadratic  Equations      .         .         .318 

XXI.  Ratio,  Proportion,  Variation 324 

XXII.  Progressions 345 

XXIII.  Permutations  and  Combinations 362 

XXIV.  Binomial  Theorem 374 

XXV.  Logarithms 380 


vii 


TEACHERS    MAY  OBTAIN   ANSWER-BOOKS, 

FOR   WHICH    NO   CHARGE   IS   MADE, 
ON    APPLICATION    TO    THE    PUBLISHERS. 


ELEMENTARY  ALGEBRA 

CHAPTER   I 

INTRODUCTION  AND   DEFINITIONS 

1.  The  science  of  number  includes  both  Arithmetic 
and  Algebra.  Algebra  may  be  defined  as  generalized 
Arithmetic. 

2.  In  arithmetic  every  number  represents  a  definite 
value.  Thus,  4  =  1  +  1  +  1  +  1.  In  algebra,  a  set  of 
symbols^  usually  letters  of  the  alphabet^  is  used  to  represent 
numbers.  A  letter  can  represent  any  number  wliatever, 
provided  its  value  does  not  change  during  a  particular 
range  of  operations. 

\ 

SYMBOLS   OF   OPERATION 

Addition  is  indicated  by  the  sign  +,  read  "plus.'' 

ihus,  4  +  1  means  the  sum  of  4  and  1 ;   a  +  d  means  the 

\  of  a  and  d, 
of 

thd^btraction  is  indicated  by  the  sign  — ,  read  "  minus, " 

"ijius,  3  —  2  means  that  2  is  to  be  subtracted  from  3  j  6  —  c 
facti^s  that  c  is  to  be  subtracted  from  b, 

1 


2  ELEMENTARY  ALGEBRA  [Ch.  I,  §  3 

Multiplication  is  indicated  by  tiie  sign  X,  and  by  the 
sign  ',  each  read  "times"  or,  "multiplied  by"  ;  and  by 
the  omission  of  sign. 

Thus,  mxn,  m'Ti,  and  mn  all  mean  the  product  of  m  and  n, 
or  of  n  and  ?/i. 

The  multiplication  sign  is  never  omitted  in  expressing 
the  product  of  numbers  in  the  form  of  digits. 
Thus,  56  indicates  50  +  6 ;  5*6  indicates  5x6. 

Division  is  indicated  by  the  signs  4-,  /,  :,  each  read 
"  divided  by  " ;  and  by  the  fractional  form. 

Thus,  a-r-b,  a/b,  a :  b,  and  -  all  indicate  the  division  of 
a  by  b. 

Equality  between  two  numbers  is  indicated,  by  the 
sign  =,  read  "is  equal  to." 

Thus,  a  =  b  indicates  that  a  is  equal  to  b. 

EXERCISE  I 

If  tt  =  1,  5  =  2,  (?  =^  3,  c?  =  4,  find  the  value  of  each  of  the 
following  : 


3. 


4. 


a  +  b 
c 

6. 

cd 

11. 

abed 
a  +  c  +  d 

b  +  d 
c 

7. 

ad 

12. 

ac  +  be  +  a     \ 
a  +  d 

c  +  d 
a 

8. 

d_b 

a      a' 

13. 

ab  +  be  +  cd 
d  +  e-b 

d—  a 

• 
e 

9. 

ad      c 
b       a' 

14. 

ae  +  ad  +  cb 
cd+\ 

c-—a 
b     • 

10. 

ab  +  be 
d      • 

15. 

ad+cd—  be 
a+  b  +  c  +  d' 

Ch.  I,  §§  4-7]     INTRODUCTION  AND  DEFINITIONS  3 

ALGEBRAIC   EXPRESSIONS 

4.  An  algebraic  expression  is  a  combination  of  number 
symbols  connected  by  any  of  the  symbols  of  operation. 

Thus,  a,  7  —  a,  6  +a-h3  +  b  are  algebraic  expressions. 

5.  A  term  of  an  algebraic  expression  is  a  combination 
of  number  symbols  not  separated  by  the  signs  +  or  — . 

Thus,  in  the  algebraic  expression  6  +  a  -^  3  —  6,  the  terms 
are  6,  a  ~  3,  and  b.  ^-^ 

6.  When  two  or  more  numbers  multiplied  together 
produce  a  certain  product,  each  of  these  numbers  is 
called  a  factor  of  the  prpduct. 

Thus,  a,  h,  and  c,  are  factors  of  abc. 

Each  of  the  factprs  of  a  number  or  the  product  of  any 
number  of  factors  is  called  a  coefl3:cient  of  the  rest  of  the 
term. 

Thus,  in  3  a,  3  is  the  coefficient  of  a ;  in  a6,  a  is  the  coefficient 
of  6 ;  in  I  ahc,  f  is  the  coefficient  of  abc,  |  a  of  he,  and  -|  ah  of  c. 

The  coefficient  is  generally  understood  to  mean  the 
number  placed  before  the  number  symbols  represented  by 
the  letters. 

If  the  coefficient  he  1,  it  is  always  omitted. 

Thus,  a  =  1  a. 

7.  The  exponent  of  a  number  is  the  symbol  in  the  form 
of  an  integer  which  represents  how  many  factors  equal  to 
the  number  affected  by  the  exponent  are  taken. 

Thus,  a^  represents  that  a  has  been  taken  three  times  as  a 
factor ;  or,  a^  =  a  •  a  •  a. 


4  ELEMENTARY   ALGEBRA  [Ch.  I,  §§  8-10 

The  exponent  affects  only  that  number  symbol  which 
it  follows,  and  at  the  upper  right  hand  of  which  it  is 
written. 

Thus,  3  a%c  means  that  a  alone  has  been  taken  twice  as  a 
factor ;  or  3  a?ho  =  3  >  a  '  a  '  b  >  c. 

If  no  number  symbol  be  written  as  the  exponent,  it  is 
always  understood  that  1  is  that  exponent. 

Thus,  in  3  a^bc,  3,  b,  and  c  are  to  be  understood  as  having  the 
exponent  1  affecting  each  of  these  numbers ;  or  3  a^bc  =  3^a-6V. 

Since  the  product  of  a  number  of  equal  factors  can  be 
called  a  power  of  that  number,  a^  can  be  read  "  a  with  the 
exponent  3"  ;  or  "a  third." 

Thus,  a'^=  a  '  a  '  a  -  a  can  be  read  ^^ a  with  the  exponent  4/' 
"  a  fourth/'  or,  '^  a  to  the  fourth  power.'' 

The  distinction  between  coefficient  and  exponent  should 
be  carefully  noticed. 

Thus,  3a=a  +  a  +  a;  and  a^  =  a  •  a  •  a. 

8.  A  monomial  is  an  expression  containing  a  single 
term. 

Thus,  2  a^,  3  b,  and  c^  are  monomials. 

9.  Similar  terms,  or  like  terms,  are  those  which  differ 
only  in  their  numerical  coefficients. 

Thus,  3  a^b,  a?b,  and  7  a?b  are  similar,  or  like,  terms, 

10.  A  polynomial  is  an  expression  containing  several 
terms. 

Thus,  2o?b  +  3  ab'^  +  5^  is  a  polynomial. 


Ch.I,§§11,12]     INTRODUCn^M'N  AND   DEFINITIONS  5 

A  polynomial  which  coiM&|^^'^  ^^^^  terms  is  called  a 
binomial ;  and  one  which  con^i^s  three  terms  is  called 
a  trinomial.  V^ 

Thus,  o?  +  6^  is  a  binomial ;  and  a^—  a?\+  h'^  is  a  trinomial. 

11.  The  positive  and  negative  terms  of  ai3L  expression 
are  those  which  are  preceded  by  the  plus  and  mkius  signs 
respectively. 

Thus,  the  positive  terms  of  a^  —  3  o?h  +  3  alP'  —  If  are  ot^ii^ 
3  a6^,  and  the  negative  terms  are  3  a%  and  h^,  ^B^ 


« 


12.  The  numerical  value  of  an  expression  is  found  by 
substituting  for  the  letters  their  values  in  numbers,  and 
performing  the  indicated  operations. 

Thus,  the  numerical  value  of  2  a,  if  a  =  4,  is  8. 


EXERCISE   II 


If  a  =  6,  5  =  4,  {?  =  3,  c?  =  2,  ^  =  1,  find   the   value   of 
each  of  the  following  expressions: 


1. 

2aJ. 

11. 

2^+3  c2. 

21. 

V,4  +  ^2J2  +  54. 

2. 

Zed. 

12. 

4^2-35^. 

22. 

a^-b^. 

3. 

4cde. 

13. 

a2-4  6'2. 

23. 

b^+c\ 

4. 

a?d. 

14. 

bad-2h\. 

24. 

b^  -  C^ 

5 

cH. 

15. 

4:a^-2Pd^. 

25. 

c^  +  cd  +  d\ 

6. 

4  aHe. 

16. 

ab+hc-}-  J2. 

26. 

c'^-cd  +  d\ 

7. 

2cH. 

17. 

2ac-c^+d^. 

27. 

2a2+52-5(?2. 

8. 

2hHdH. 

18. 

a^+ab  +  P. 

28. 

j2_4j  +  4. 

9. 

6  cd^e\ 

19. 

a^-2ab-\-b^. 

29. 

2a%'^cdh.  . 

10. 

7  ahcdh. 

20. 

a^  +  J3. 

30. 

^3-^2^  + 3  ^^2. 

g  ELEMENTARY  AlGEBRA  [Ch.  I,  §§  13, 14 

13.  Aggregation,  the  proces^'^  of  taking  the  result  of  several 
operatio7is  as  a  whole^  is  ijj^icated  by  the  symbols  (  ),  {  }, 
[  ],  read  respectively.^' parenthesis,"  "brace,"  "bracket." 

Thus,  aQ)-\-c),  a^pj^c],  a\h-\-c\  all  mean  that  the  sum  of 
h  and  c  is  to  be  ^ijultiplied  by  a. 

ORDER   OF   OPERATIONS 

14.  In  any  polynomial  in  which  the  various  signs  of 
operation  occur,  the  plus  and  minus  signs  are  used  to 
separate  terms. 

The  operations  of  multiplication  and  of  division  are  to  he 
performed  before  those  of  addition  and  subtraction. 

Thus,  28  -T-  4  —  2  X  3  contains  two  terms,  a  plus  sign  being 
imderstood  as  preceding  28 ;  +  28  -^  4  —  2  x  3  =  first  term 
(28  --  4)  -  the  second  term  (2x3); 

28--4-2x3  =  (28--4)-(2x3)  =  7-6  =  l.  ' 

Were  this  problem  to  be  given  orall}/  .r.  arithmetic,  it  might  be 
understood:  28'--4  =  7;  7-2  =  5;  5x3  =  15. 

The  difference  between  the  algebraic  usage  and  the 
arithmetical  oral  statement  is  to  be  carefully  noticed. 

EXERCISE  III 

If  a  =  1,  5  =  2,  (?  =  3,  cZ  =  4,  find  the  value  of  the  follow- 
ing expressions : 

1.  a-{-d^b.  6.  b(d-ay, 

2.  2b^xc-2ab.  7.  {Sa-b)(Sa  +  b}. 

3.  iaW-exd,  8.  (b  +  a)^ -r-^d  -  a). 

4.  Bac'^d-^d'^+Sb^.  9.  Sa^d^9bc  +  b^. 

5.  (3a  +  2^)-lla  +  J2.  10,  4ax52^2a3^  +  6V. 


Ch.  I,  §  15]        INTRODUCTION  AND   DEFINITIONS  7 

USE   OF  LITERAL  NOTATION 

15.  The  properties  of  numbers,  whether  expressed  by 
integers  or  by  letters,  are  identical. 

The  advantage,  therefore,  of  representing  numbers  by 
letters  lies  in  the  fact  that  the  letter,  being  a  general 
number,  often  leads  to  a  general  conclusion,  expressed  as  a 
formula.  In  arithmetic  the  principle  is  taught  that  inter- 
est =  principal  x  time  x  rate  per  cent ;  or  that  i  —  prt^ 
whatever  may  be  the  numerical  values  of  the  letters. 

Moreover,  literal  notation  is  often  advantageously  used 
as  a  sort  of  shorthand.  For  example,  four  times  a  cer- 
tain number  equals  the  sum  of  60  and  three  times  that 
number.     Expressing  the  problem  in  arithmetic, 

4  times  the  number  =  60  +  3  times  the  number. 

Expressing  the  same  problem  in  algebraic  language, 
taking  x  to  represent  the  number, 

4a;=60  +  3a;. 

The  advantage  of  the  algebraic  form  of  statement  lies 
in  the  fact  that  it  is  merely  a  statement  in  shorthand, 
where  x  takes  the  place  of  the  printed  words  "  the  number." 

EXERCISE  IV 

1.  Express  in  algebraic  form  the  sum  of  twice  a  num- 
ber, a,  and  three  times  that  number  ;  the  product  of  five 
times  a  number  and  four  times  that  number. 

2.  If  1  barrel  of  flour  costs  i5,  how  much  will  2  barrels 
cost?     3  barrels?     a  barrels?     h  barrels? 


8  ELEMENTARY  ALGEBRA  [Ch.  I,  §  15 

3.  If  20  barrels  of  flour  cost  $  80,  what  will  be  the  cost 
of  1  barrel?  If  a  barrels  cost  $80,  what  will  be  the  cost 
of  1  barrel  ? 

4.  If  a  man  earns  15  a  day,  how  much  will  he  earn  in 
4  days  ?     in  h  days  ?     in  c  days  ? 

5.  The  sum  of  two  numbers  is  20.  If  one  of  the  num- 
bers is  8,  what  is  the  other  number  ?  If  one  of  the  num- 
bers is  a,  what  is  the  other  number  ? 

6.  If  one  part  of  8  is  6,  what  is  the  other  part  ? 

7.  If  one  part  of  a  is  2,  what  is  the  other  part  ? 

8.  If  one  part  of  2  is  a,  what  is  the  other  part  ? 

9.  If  one  part  of  a  is  x^  what  is  the  other  part  ? 

10.  If  one  part  of  x  is  6,  what  is  the  other  part  ? 

11.  What  is  the  product  of  two  numbers,  if  one  factor 
is  a  and  the  other  b  ? 

12.  What  is  the  divisor,  if  the  dividend  is  27  and  the 
quotient  3  ?     If  the  quotient  is  a  ? 

13.  The  divisor  of  a  certain  number  is  a  and  the  quo- 
tient be     What  is  the  dividend  ? 

—  14.    How  much  is  8  increased  by  3  ?     8  increased  hj  a? 
a  decreased  by  4  ?     m  decreased  by  2  a;  ? 

15.  By  how  much  does  12  exceed  8  ?     12  exceed  a  "i 
a  exceed  12?     a  exceed  x? 

16.  What  is  the  excess  of  20  over  11  ?     of  20  over  x  ? 
of  X  over  20  ?  oi  x  over  i/  ? 

-^    17.   What  is  the  quotient  of  20  divided  by  the  excess  of 
X  over  200  ? 

18.   If  X  is  the  smaller  part  of  5,  what  is  the  larger  part  ? 


Ch.  I,  §  15]        INTRODUCTION  AND   DEFINITIONS  9 

19.  If  10  is  the  larger  part  of  x,  what  is  the  smaller 
part  ? 

20.  How  much  does  8  lack  of  13  ?  of  a  ? 

21.  How  much  does  a  lack  oi  x?  of  22  ? 

22.  How  much  does  x  lack  of  13  ?  of  m  ? 

23.  If  A  is  30  years  old  now,  how  old  will  he  be  in 
4  years  ?  in  x  years  ? 

24.  If  A  is  now  a  years  old,  what  would  half  his  age 
be  ?  three  times  his  age  ? 

25.  If  A  is  18  years  old  now,  how  old  was  he  4  years 
ago  ?  a  years  ago  ? 

26.  If  A  is  25  years  old  now,  what  was  three  times  his 
age  a  years  ago  ? 

27.  What  is  the  average  age  of  two  men,  the  age  of  the 
first  being  30,  and  the  second  being  a  ? 

^     28.    If  3  is  the  tens'  digit  of  a  number  of  two  digits,  and 
a  the  units'  digit,  what  is  the  number  ? 

^   29.    If  a  is  the  greater  part  of  a  number,  and  the  differ- 
ence between  the  parts  is  4,  what  is  the  other  part  ? 
^  30.    If  a  is  the  smaller  part  of  a  number,  and  if  the 
smaller  part  lacks  4  of  the  larger  part,  what  is  the  larger 
part? 

31.  If  2  a  +  3  represents  a  certain  number,  what  repre- 
sents a  fourth  of  that  number  ? 

32.  By  how  much  does  three  times  a  exceed  22  ? 

33.  By  how  much  is  the  third  part  of  a  below  9  ? 

34.  If  A  has  X  dollars,  B  twice  as  much  as  A,  and  C 
as  much  as  A  and  B  together,  how  much  has  B  ?  how 
much  has  C  ? 


10  *        ELEMENTARY  ALGEBRA         [Ch.  I,  §§  16,  17 


POSITIVE   AND  NEGATIVE   NUMBERS 

16.  Up  to  this  time  the  restriction  has  always  been 
made  that  the  quantity  to  be  subtracted,  the  subtrahend, 
must  be  less  than  the  quantity,  the  minuend,  from  which 
the  subtrahend  is  to  be  subtracted.  Since  7  is  greater 
than  4,  it  is  possible  to  subtract  4  from  7.  Expressed 
in  arithmetical  language,  7  —  4  =  3.  Since  4  is  less  than 
7,  it  is  not  possible  to  subtract  7  from  4.  But  there  is  a 
mathematical  necessity  for  making  the  process  of  subtrac- 
tion always  possible. 

17.  It  is  evident  that  a  new  sort  of  number  must  be 
employed  if  subtractions  are  always  possible.  Numbers 
hitherto  employed  can  be  represented  as  shown  in  Fig- 
ure 1. 

012345678 

■  : 1 I I L \ 1 1 I I 

Fig.  1. 

If  a  straight  line  of  indefinite  length  is  divided  into 
units  of  length  from  zero,  the  natural  numbers  can  be 
represented  by  successive  repetition  of  this  unit  of  length 
in  a  direction  extending  indefinitely  towards  the  right. 
These  numbers  will  be  seen  to  increase  by  a  unit,  count- 
ing from  left  to  right ;  and  to  decrease  by  a  unit,  count- 
ing from  right  to  left.  The  addition  of  2  and  3  can  be 
illustrated  by  counting  from  zero,  two  units  towards  the 
right,  and  then  by  counting  three  more  units  from  2 
towards  the  right.  The  subtraction  of  2  from  3  can  be 
illustrated  by  counting  three  units  from  zero  towards  the 
right,  and  then  by  counting  two  units  from  3  in  the  op- 
posite direction  towards  the  left.     If,  however,  the  prob- 


Ch.  I,  §§  18,  19]     INTRODUCTION  AND   DEFINITIONS  11 

lem  were  to  subtract  a  greater  from  a  lesser  number,  — 
for  example,  to  subtract  3  from  2,  —  the  process  is:  count 
from  zero  two  units  towards  the  right ;  try  to  count  three 
units  from  2  towards  the  left ;  two  units  can  be  counted 
up  to  zero ;  the  third  unit  will  seem  to  be  beyond  zero  to 
the  left.  It  is  evident  that  the  counting  cannot  continue 
further  unless  there-  are  ne\r  units  which  are  different  in 
character  towards  the  left  of  zero. 

18.  An  abstract  number  is  used  without  application  to 
things,  as  3,  4,  6 ;  a  concrete  number  is  used  with  applica- 
tion to  things,  as  3  men,  4  inches,  6  cubic  feet.  Concrete 
numbers,  or  quantities,  are  often  opposite  in  character. 
The  following  are  examples  of  opposite  concrete  quanti- 
ties: $20  gain  and  $15  loss;  2  inches  to  the  right  and  4 
inches  to  the  left ;  10  degrees  above  zero  and  5  degrees 
below  zero ;  25  degrees  north  latitude  and  4  degrees  south 
latitude.  If  two  concrete  quantities  of  opposite  kinds  be 
combined,  the  effect  of  one  is  to  decrease,  destroy,  or  to 
reverse  the  state  of  the  other.  For  example:  $20  gain 
combined  with  $15  loss  destroys  the  loss  of  $15  and  leaves 
a  gain  of  $5. 

19.  Differences  that  arise  from  subtracting  quantities 
from  lesser  quantities  are  called  negative  quantities. 
Quantities  that  are  not  negative  are  called  positive  quan- 
tities. Positive  quantities  are  represented  thus  :  -f-  3,  +  5; 
while  negative  quantities  are  represented  thus :  —3,-5. 
The  former  are  read :  "positive  3,"  "positive  5  "  ;  the  latter 
are  read:  "minus  (negative)  3,"  "minus  (negative)  5." 
The  signs  -f  and  —  are  also  used  to  indicate  the  processes 
of  addition  and  of  subtraction.  Therefore,  for  the  present, 
positive  numbers  will  be  indicated  thus:  (+3),  (+5); 
and  minus  (negative)  numbers  thus:   (—3),  (—5). 


12  ELEMENTARY   ALGEBRA         [Ch.  I,  §§20-22 

20.    The  series  of  positive  and  negative  numbers  can  be 
represented  as  shown  in  Figure  2 : 


-9-8-7-6-5-U-3-3-1    0123U56789 
I      I      I      I     I      I      I      I      I,     I      I     I      I      i      I      I      I      I      I 

Fig.  2. 


Numbers  passing  from  zero  in  the  positive  direction 
increase  indefinitely,  and  numbers  passing  from  zero  in 
the  negative  direction  diminish  indefinitely.  Positive  and 
negative  numbers  taken  together  are  called  algebraic  num- 
bers. The  sign  +,  indicating  a  positive  number,  is  some- 
times omitted ;  the  sign  — ,  indicating  negative  numbers,  is 
never  omitted.  When  no  sign  is  written  before  a  number, 
the  plus  sign  is  always  understood. 

21.  The  absolute  or  numerical  value  of  a  number  de- 
pends upon  the  number  of  units  contained  in  the  number, 
no  reference  being  paid  to  its  sign,  or  its  quality  of  oppo- 
sition, that  is,  its  direction  towards  the  right  or  towards 
the  left.  For  example  :  (  +  7)  and  (  —  7)  are  equal  in 
absolute  or  numerical  value. 

22.  A  negative  number  may  be  considered  as  indicating 
a  delayed  or  postponed  subtraction.  For  example  :  (—1), 
since  it  is  a  difference  obtained  by  subtracting  a  quantity 
one  unit  greater  than  a  second  quantity,  indicates  that 
(+1)  still  remains  to  be  subtracted.  Since  the  addition 
of  (  —  1)  to  a  second  number  means  the  subtraction  of 
(+1)  from  the  second  number,  by  applying  the  same 
principle  to  any  negative  number,  it  is  evident  that  add- 
ing a  negative  number  to  a  second  number  is  equivalent  to 
subtracting  a  positive  numl?er  {of  the  same  absolute  value  as 
the  negative  number)  from  the  second  number. 


Ch.  I,  §  23]  INTRODUCTION   AND   DEFINITIONS  13 

23.    1.    Add  (+3)  and  (+5). 

-9'8-7-6'5-U-3-2-l    0123U56789 
I      I     I      I      I      I      I      I      i     t     I      I     .\     I      I      I      I      I      I 

Fig.  2. 

The  sum  of  (+3)  and  (+5)  is  found  by  counting, 
from  (+3),  five  units  in  the  positive  direction;  and  is, 
therefore,  (  +  8). 

2.  Add  (-3)  and  (-5). 

The  sum  of  (—  3)  and  (—5)  is  found  by  counting,  from 
(  —  3),  five  units  in  the  negative  direction ;  and  is,  there- 
fore, (—8). 

3.  Add  (+5)  and  (-3). 

The  sum  of  (+  5)  and  (~  3)  is  found  by  counting,  from 
(  +  5),  three  units  in  the  negative  direction  ;  and  is,  there- 
fere,  (+2). 

4.  Add    (-5)  and  (+3). 

The  sum  of  (  —  5)  and  (  +  3)  is  found  by  counting,  from 
(  —  5),  three  units  in  the  positive  direction;  and  is, 
therefore,  (  —  2). 

If  a  and  b  represent  any  two  integers,  positive  or  nega- 
tive, 

(  — a)  +  (  — J)=— a— J, 
i  +  a}  +  (-b}=+a-b, 
(-^)  +  (  +  J)  =  -a  +  6. 

Zero  may  be  defined  as  the  sum  of  that  positive  and  that 
negative  number  tvhich  are  equal  in  absolute  value. 


14  ELEMENTARY  ALGEBRA  [Ch.  I,  §  24 

RULE  FOR  ADDITION  OF  TWO  NUMBERS 

If  both  numbers  are  positive^  the  sum  will  be  positive  and^ 
equal  to  the  sum  of  the  absolute  values  of  the  numbers.  If 
both  numbers  are  negative^  the  sum  will  be  negative  and  equal 
to  the  sum  of  the  absolute  values  of  the  numbers.  If  one 
number  is  positive  and  the  other  negative^  the  absolute  value 
of  the  sum  tvill  be  the  difference  of  the  absolute  values  of  the 
numbers^  and  will  be  positive  or  negative  according  as  the 
number  of  greater  absolute  value  is  positive  or  negative. 

24.  Two  operations  are  said  to  be  inverse  to  each  other 
when  the  effect  of  one  is  to  undo  the  other. 

Subtraction  is  the  inverse  operation  to  addition  ;  and  may 
be  defined  as  the  process  oi  finding  from  two  given  numbers 
a  third  number  so  that  the  sum  of  the  first  and  the  third  is 
equal  to  the  second. 

The  process  of  subtraction  depends  upon  the  principle 
in  §  22. 

1.  Subtract  (  +  3)  from  (  +  5). 

I     j     I      I      I     I     J     I      I      I      I      I      I      I      I      I      I      I      I 

Fio.2.-  ^ 

The  result  of  subtracting  (  +  3)  from  (  +  5)  is  found  by- 
counting,  from  (  +  5),  three  units  in  the  negative  direc- 
tion ;  and  is,  therefore,  (  +  2). 

2.  Subtract  (  —  3)  from  (  —  5). 

The  result  of  subtracting  (  —  3)  from  (  —  5)  is  found  by 
counting,  from  (  —  5),  three  units  in  the  positive  direc- 
tion ;  and  is,  therefore,  (  —  2). 

Three  units  are  counted  from  (  —  5)  in  the  positive  direc- 
tion because  the  subtraction  of  a  negative  quantity  is  equiva- 
lent to  the  addition  of  its  absolute  value. 


Ch.  I,  §  24] 


INTRODUCTION  AND   DEFINITIONS 


15 


3.  Subtract  (  —  3)  from  (  +  5). 

The  result  of  subtracting  (  —  3)  from  (  +  5)  is  found  by- 
counting,  from  (  +  5),  three  units  in  the  positive  direc- 
tion ;  and  is,  there foi^,  (  +  8). 

4.  Subtract  (  +  3)  from  (-5). 

The  result  of  subtracting  (  +  3)  from  (  —  5)  is  found  by 
counting,  from  (  —  5),. three  units  in  the  negative  direc- 
tion ;  and  is,  therefore,  (  —  8). 

If  a  and  h  represent  any  two  integers,  positive  or  nega- 
tive, 

(i  +  a)-(  +  h^=+a-h, 
(  --  ^)  —  (  —  J)  =  —  a  +  J, 
l  +  a)-l-h)=  +a  +  h, 

Rule  for  Subtraction  of  Two  Numbers :  Change  the  sign 
of  the  subtrahend  and  add  the  result  to  the  minuend. 


EXERCISE  V 


Find  the  values  of  the  following  indicated  operations 


1.  (+   3)  +  (+   5) 

2.  (-   3)-(-   5) 

3.  (+   3)  +  (-   5) 

4-  (-   5)  +  (+   3) 

5-  (+    T)-(+   4) 

6.  (+   6)-(+    7) 

7.  (+    9).- (-12) 

8.  (+12) -(-15) 

9.  (+    3)-(-    7) 
10.  (-   !)-(+   4) 

21 


11. 
12. 
13. 
14. 
15. 
16. 
17. 
18. 
19. 
20. 


(+    8)-(- 


(+  4)-(+   5). 

(+  6)- (  +  7). 

(+  4) +  (+5). 

(-  6) -(-5). 

(-f-  4) +  (-4). 

(+  4) -(+4). 

(-  7) +  (-4). 

(+  8) -(+5). 

(+  7) -(+8). 
(4-12)-(-4). 
9). 


16  ELEMENTARY   ALGEBRA         [Ch.  I,  §§  25,  26 

25.  The  product  of  two  algebraic  numbers  is  a  third  num- 
ber whose  absolute  value  is  the  product  of  the  absolute  values 
of  the  two  numbers  ;  and  is  (1)  positive  if  both  numbers  are 
each  positive  or  negative^  and  negative  (2)  if  one  of  the 
numbers  is  positive  and  the  other  negative. 

The  operation  of  finding  the  product  of  two  numbers  is 
called  multiplication.  To  find  the  product  of  a  and  b  is 
to  multiply  a  and  5,  or  to  multiply  b  and  a.  The  product 
of  a  and  b  is  indicated  thus  :   (a  x  S),  or  (aJ),  or  ab. 

Since  the  arithmetical  product  of  the  absolute  values  of 
the  factors  is  not  determined  by  the  order  of  the  factors, 
by  definition  the  product  of  a  and  b  is  the  same  as  the 
product  of  b  and  a.  If  ab  indicates  the  product  of  a  and 
6,  and  ba  indicates  the  product  of  b  and  a,  ab  =  ba. 

(+5)x(+3)=:(+15), 

(-5)x(-3)  =  (  +  15), 

(+5)x(-3)  =  (-15), 

(~5)x(+3)  =  (-15). 
In  general,       (+^)  x  (+J)  =  (+  ^5), 

I- a]  X  (-S)  =  (+aS), 

(+a)  x(-J)  =  (-a5), 

(-a)  X  (+6)  =  (-a6). 

The  Law  of  Signs  in  Multiplication  :  Like  signs  give  posi- 
tive^ and  unlike  signs  give  negative  products. 

26.  The  absolute  value  of  the  quotient  of  two  numbers  is 
the  quotient  of  the  absolute  values  of  the  numbers;  and  is 
(1)  positive  if  both  numbers  are  each  positive  or  negative^ 
and  is  (2)  negative  if  one  of  the  numbers  is  positive  and  the 
other  negative. 

The  operation  of  finding  the  quotient  of  two  numbers  is 
called  division.  Division  is  tlie  operation  inverse  to 
multiplication. 


Ch.  I,  §  26] 


INTRODUCTION  AND   DEFINITIONS 


vl^ 


Since,  §  25, 

(+5)x(+3)  =  (+15),  (+15) 


(_5)x(-3)  =  (+15),  (  +  15)^(-3)  =  (-5) 


(  +  5)x(-3)=(- 
(_5)x(+3)  =  (- 


15),  (-15) 


(+3)=  (+5) 


(_3)  =  (+5) 


15),  (-15)^(+3)  =  (-5). 


In  general. 


(+aJ)^(+5)  =  (+a), 


C-ab) 


C-ab) 


(_6)  =  (+a), 


(+a5)^(-6)  =  (-a). 


(+5)  =  (-5).<- 


The  Law  of  Signs  in  Division  is  :  Like  signs  give  positive^ 
and  unlike  signs  give  negative  quotients. 


EXERCISE   VI 


Find  the  values  of  the  following  indicated  operations: 


1.  (  +  3)(-2). 

2.  (-4)(-5). 

3.  (-8)(-3> 

4.  (-9)(-4). 

5.  (  +  6)(-4). 

6.  (-7)(+3). 

7.  (-5)(-6). 

8.  (-8)(+3). 

9.  (-9)(-5). 


10. 

li. 

12. 
13. 
14. 
15. 
16. 
17. 
18. 


(+6)(  +  7). 

(_9)-H(+3). 

(_8)^(+4). 

(+10)^(+5). 

(_10)^(  +  2). 

(_12)^(-4). 


(+12) 
(+15) 
(-16) 


(-12). 

(+3). 
(-8). 


18  ELEMENTARY   ALGEBRA  [Ch.  I,  §  27 

27.  The  sign  +  may  be  used,  §  3,  to  denote  addition^ 
and,  §  19,  to  indicate  positive  numbers.  In  practice,  how- 
ever, the  sign  +  is  omitted  in  indicating  positive  numbers. 
Thus,  (  +  4),  4,  are  identical.  Henceforth,  in  this  book, 
positive  numbers  will  be  represented  by  the  absence  of 
sign.  Thus,  4  means  positive  4,  and  +4  means  the  addi- 
tion of  positive  4. 

The  sign  —  may  be  used,  §  3,  to  denote  subtraction, 
and,  §  19,  to  indicate  negative  numbers.  In  conformity 
with  general  usage,  negative  numbers  will  be  henceforth 
represented  by  numbers  .preceded  by  the  sign  — .  Thus, 
(  —  5)  and  —5  are  identical.  The  sign  — ,  denoting  a 
negative  number,  is  never  omitted. 


EXERCISE   VI] 

[ 

Simplify  the  following: 

1.    (4)  +  (3). 

16. 

(-5) +.(-7). 

2.    (4) -(3). 

17. 

(4)  .  (-  3). 

3.    (4) +  (-3). 

18. 

(-5).  (2). 

4.    (4) -(-3). 

19. 

(-  6)  •  (-  5). 

5.    (-4) +  3. 

20. 

(-4).  (4). 

6.    4  +  3. 

21. 

(-9).  4. 

7.   4-3. 

22. 

(_12).(-3). 

8.    4 -(-3). 

23. 

8-5. 

9.    4 +  (-3). 

24. 

-12-3. 

10.    8  +  (-2). 

25. 

24  -  (-  3). 

11.    (-2) +8. 

26. 

_  36  ^  (-  6). 

12.    7  -  (-  5). 

27. 

-  54  ^  18. 

13.    7 +  (5). 

28. 

_  39  ^  (_  13). 

14.    7  +  5. 

29. 

-  65  H-  5. 

15.    7 +  (-5). 

30. 

50  H-  (  -  25). 

CHAPTER  II 

ADDITION  AND  SUBTRACTION 

28.  The  addition  of  two  numbers,  or  quantities,  whether 
positive  or  negative,  has  already  been  illustrated,  and  the 
rule' given,  in  §  23. 

The  sum  of  three  quantities  is  the  sum  of  the  first  two 
quantities  and  the  third  quantity  ;  similarly,  the  sum  of 
four  quantities  is  the  sum  of  the  first  three  and  the  fourth 
quantity. 

Thu^2  a'  +  3bc)\-c'  +  m  =  (2  a'  +  3  6c)  +  c^  +  m 
=  [(2a2  +  3&c)+c2]+m. 

29.  Addition  is  subject  to  two  laws  (whose  truth  is  as- 
sumed), the  first  of  which  is  the  Commutative  Law,  —  the 
sum  of  two  or  more  numbers  is  independent  of  the  order  in 
■which  the  addition  is  performed. 

Thus,  4  +  5  =  5  +  4;  or,  in  general,  a  +  h^h  +  a. 

I|  Addition  is  also  subject  to  the  Associative  Law,  —  the 
sum  of  three  or  more  numbers  is  independent  of  the  way  in^ 
which  successive  terms  are  grouped  in  the  process  of  addi- 
tion. 

Thus,  4  +  5  +  2  =  (4 +  5) +2  =  9  +  2  =  11, 

and  4  +  5  +  2  =  4  +  (5  +  2)  =  4  +  7  =  11, 

in  general,       a  +  h+  c^{a  +  h)  +  c  =  a  +  (6  +  c). 

39 


20  ELEMENTARY   ALGEBRA  [Ch.  II,  §  30 

The  Associative  Law  gives  a  short  method  for  combin- 
ing positive  and  negative  terms. 

Thus,  the  sum  of  the  positive  terms  of  22  —  11  +  12  —  5  + 
6  —  17  is  40 ;  the  sum  of  the  negative  terms  of  22  —  11  +  12 
_  5  +  6  -  17  is  -  33.  Hence  22  - 11  + 12  -  5  +  6  - 17  =  (22 
+ 12  +  6)  +  (  - 11  -  5  - 17)  -  40  -  33  =  7. 

EXERCISE  VIII 

Find  the  sum  of  the  following  numbers : 

1.  20-3  +  7-8.  6.    30-14-16  +  5. 

2.  16-22  +  12-5.  7.   27-18-17  +  8. 

3.  1_  12 +  13 -7.  8.   6-22  +  33  +  12-6. 

4.  8-9-10-11.  9.    24-8-13  +  7  +  5-16. 

5.  20-14-13  +  27.       10.   6  +  8-21  +  17-8-5-13.. 

11.  _  6  + 5- 19 +  13- 20 +  4+ 7. 

12.  9-8  +  15  +  3-19-11-6. 

13.  12-8-7-14  +  15-13  +  20  +  5. 

ADDITION   OF  LIKE   TERMS 

30.  Like  terms  can  be  combined  into  a  single  term. 
Just  as  in  arithmetic,  the  sum  of  4  bushels  and  3  bushels 
is  indicated  by  4  bu.  +  3  bu.  =  7  bu.,  so,  in  algebra, 
Sa?b  +  5  a%  =  8  a%.  Hence,  to  add  like  terms,  add  their 
numerical  coefficients^  and  prefix  this  sum  as  the  numerical 
factor  of  the  literal  part. 

Thus,  3  a^^  +  5  a%  +  2  a^^  =  10  a% 

and  — 2a— 3a  —  5a=  —  10  a, 

and  262_362  +  1062  =  96l 


Ch.  II,  §  30]  ADDITION  AND   SUBTRACTION  21 

EXERCISE   IX 

Find  the  sum  of  the  like  terms  in  the  following : 

1.  2a  +  Sa  —  ia, 

2.  6w+5m  —  7m. 

3.  2c-Sc-4:C.  ^ 

4.  7a2-4a2  +  2a2. 

5.  8ab-2ab -Sab +  12  ab. 

6.  Q  x^  —  "ii  x^  +  b  x^  —  x^, 

7.  llxy  —  1  xy  +  4Lxy  —  ?>  xy. 

8.  12bc-?>bc  +  Qbc-2bc. 

9.  —  x^y  —  2  x^y  —  3  x^y  +  5  ^r^y. 

10.  10  b^e-Sb^c-5b'^c  +  4.b'^c. 

11.  2  62^2  __  3  J2^  _  7  J2^2  _  6  J2^2.    -  -    ^V4^  ^  V      * 

12.  4iab  — 5  ab +  7  ab  — 11  ab —  12  ab. 

13.  5  m7^  —  4  mn  —  6  m?^^  —  7  m/z.  —  m7^  +  2  m^. 

14.  6  a5  —  7  aJ  —  2  aJ  —  a5  +  12  a5  +  22  ab. 

15.  a:2-lla;2-13ii;2+7^2_5^2_4^2^7^2_9^. 

16.  mn  +  2  mn  —  3  mn  —  7  m7^  +  13  mn  —  14  mn. 

17.  -  a6  +  7  a6  -  13  aJ  +  12  (26  -  7  aS  -  15  aJ. 

18.  :i:2_3^2_4^2+7^2_9^2_ll^_4^2+5^^ 

19.  ?/2-ll2/2_13^2  +  5^2_4^2^3^2_223/2. 

20.  a2-3a2  +  4a2-6a2-7a2-32a2  +  50a2. 

21.  —  aS  +  4  a6  —  7  ah  +  b  ab  —  1'^  ab  +  11  ab  —  56  ab. 

22.  a -17  a +  33  a -44 a +  109 a- 64 a +  32 a. 

23.  :i;2y  -  3  :r2t/  +  5  rr2^  +  22  :?:22/  -  17  x'^y  +  37  x^y. 

24.  -  17  62  -  33  62  +  105  62  +  62  62  -  109  62  -  56  62. 

25.  6  a6  -  17  a6  +  33  ab  -  512  ab  +  203  ab  +  1002  ab. 


22  ELEMENTARY   ALGEBRA  [Ch.  II,  §  31 

ADDITION   OF  POLYNOMIALS 

31.  Let  A=h+c+d,  and  let  U  =  m  —  n—p.  The 
addition  of  these  two  polynomials  is  indicated  thus: 
A  +  U=  (h  +  c  +  d)  +  (jii  —  n—p}»  (1)  The  parenthesis 
may  be  dropped,  and  the  equivalent  expression  may  be 
written:  A  +  U=b  +  c  +  d  +  m  —  n—p,   (2)     Expression 

(1)  indicates  the  sum  of  the  numerical  values  of  the  poly- 
nomials b+c  +  d  and  m  —  n—p;  the  numerical  value  of 
expression  (2)  is  independent  of  the  order  of  the  terms, 
and  may  be  considered  as  the  sum  of  the  numerical  values 
of  the  first  three,  and  the  last  three  terms,  which  is  exactly 
the  result  of  expression  (1).     Hence  expressions  (1)  and 

(2)  are  equivalent.  Whence  is  the  following  rule  for  the 
addition  of  polynomials  :  Write  the  polynomials  in  order ^ 
retaining  the  sign  of  each  term. 

If  the  polynomials  contain  like  terms,  these  terms  should 

be  united. 

1.    Add  ^2  +  2  a5  +  52  and  a^-2ah  +  h^. 

The  work  will   be  simplified  by  arranging  like  terms 

under  like  terms  before  combining. 


2  a'  +252 

If  the  sum^  of  more  than  two  polynomials  is  required, 
the  process  is  similar. 

2.    Add  a2  -  3  ah,  6  a5  -  b^  11  a^  +  3  ab-  12  b^. 

a'  —  3ab 

6ab-^  52 
11  a^ -{-Sab -12  b^ 
12a'  +  6ab~13b' 


Ch.  II,  §§  32, 33]     ADDITION  AND   SUBTRACTION  23 

32.*  The  process  of  finding  the  sum  of  several  poly- 
nomials containing  like  terms,  may  be  still  further  abridged 
by  the  method  of  Detached  Coefficients;  that  is,  by  omitting 
the  literal  parts  of  several  like  terms. 

Thus,  in  finding  the  sum  oi2m  +  3n  —  6p,  Urn  — An -\- 2 p, 
and  —7m  +  n—p,  omit  all  literal  factors  except  in  the  .first 
line  and  arrange  the  terms  thus : 

11     _4     +2 
-7     +1     -1 
6m  —5p 

The  advantage  of  this  method  is  simply  in  the  labor 
saved  by  omitting  the  literal  factors. 

CHECKS  FOR   OPERATIONS 

33.  It  is  often  useful  to  test,  or  check,  the  results  ob- 
tained in  the  processes  of  addition,  subtraction,  multipli- 
cation, and  division  with  the  results  obtained  in  the  same 
operations  obtained  from  the  numerical  values. 

1.  (3a^-2x^  +  5x  +  l)  +  (3x'-2x  +  3)=.3x^  +  x^+3x  +  4:. 
In  each  of  the  above  expressions  take  aj  =  1,  then 

(3-2 +  5  +  1) +  (3- 2 +3)  =  3  +  1 +  3  +  4, 
or,  7+4  =11, 

11  =11. 

2.  (5  a^-^  6'2;2-  3  a-b  +  m)  +  (2  6^  +  3a-6  -  4  m)  =  5  a^ 

-^4.l^-3m. 
In  each  of  the  above  expressions  take  a  =  6  =  m  =  1. 
(5  ...6 -3+1) +  (2 +  3 -4)  =  5- 4^3, 
or,  *  -3+  l  =  -2, 

-  2  =  -  2. 


24  ELEMENTARY  ALGEBRA  [Ch.  L:,  §  33 

EXERCISE  X 

Add  the  following  polynomials : 

1.  2a  +  3h  +  2c,  a+h  —  c. 

2.  8h+  2c-d,  2b  —  5c  —  d. 
Z,  m-\-n—l p^   —m  — n  +  1  p, 

4.  ah  +  hc  +  alP'c^  2  ah  —  8  he  —  4:  ah^o. 

5.  2ah  +  a^  +  3h^  2ab  +  Sa^-4:  h\ 

6.  4:xy  —  x^  +  4:  y\  6  x^  —  5  x^  —  1  y^. 

7.  8a^-bah-\-lG\  2^2-6^5  +  4^2. 

8.  x^  —  xy  ■\-  y^^  8x^  —  6  xy  —  4:  y^. 

9.  Im— 2n-{-p  +  6q^  6  m  + 5n  — 6  p +  2  q. 

10.  4:x  —  2y  +  Sz—8,  Qx-dy  —  Sz-^G. 

11.  a2  +  52,  a2-3a6  +  J2,  2ah-2h^. 

12.  m^  +  mn  +p^  3  m2  —  2  mn  —  jt?2,  6  m2  —  3  ^n  +  2  jt?2^ 

13.  5  m  —  10  n  +  /ip,  m  —  7  ^  +  njt?,  6  m  +  12  n  —  4  7^p. 

14.  x^  —  xy  +  2/2,  a;2  +  2  ^?/  +  2/2,   —  a;2  —  4  rry  —  4  ?/2. 

15.  12a2-lla6  +  6?)2,  -5^2  +  2^6-3 J2,  6a2  +  8a5  +  4J2.: 

16.  6  m2  —  3  07171  +  5  7i2,   5  nfi  +  8  mn  —  4  7^2^ 

—  10  ^2  +  5  mn  +  12  n^, 

17.  a6  —  ac?  +  acZ,  ac  —  ah  +  ad^  ad—  ac+  ah. 

18.  rrfi  —  rfi  +  jt?2,  n^  —  rrfi—  p^^  p'^  —  nfl—  r?. 

19.  a^-ah  +  h\  h'^  -  a^ -^  ah,  K- -^  ah  -  d^. 

20.  2a-3c2  +  4(^,  J2_3^2_|.2cZ,  h'^-a-2a\ 

21.  5  a;2  —  11  ^z:?/  +  12  2/^  x^y'^  —?fxy-\-  2/2,  a;2  —  2/2. 

22.  12  62__i0J^  +  15c;,  a2-10  62  +  ll5^,  d-Uh^-^lla'^ 

23.  22  2:2-3%  +  42/2,  15  52/-4  2/2-2a;2,  22hy-~y^  +  9x^ 

24.  6a26-7a2(?-5e2^+8  52a,  11  c^+S  (72^  +  6  ti2^-9  ^2^ 

25.  9x32/-a;4-12ii:2^2_i4^^3_,.y^  a^^- 6  ii^^j/ +  10  ^^ 

^2yK 


Ch.  II,  §§  34,  35]    ADDITION  AND   SUBTRACTION  25 

SUBTRACTION 

34.  The  subtraction  of  two  quantities  has  already  been 
defined,  and  the  rule  given  in  §  24. 

SUBTRACTION   OF  LIKE   TERMS 

35.  Just  as  in  arithmetic  the  process  of  subtracting  3 
barrels  from  4  barrels  is  indicated  by  4  bbls.  —  3  bbls.  =  1 
bbl.,  so,  in  algebra,  the  subtraction  of  3  a  from  4  a  is  indi- 
cated i  a  —  S  a  =  a.  But,  §  19,  4  a  can  be  subtracted  from 
3  a,  and  is  indicated  8  a  —  4:  a=  — a;  that  is,  —  a  must  evi- 
dently be  added  to  4  a  to  make  3  a. 

Similarly,  2a— (  —  5  a)  =  7  a i  —5a—  (6  a)=  —11  a. 

In  §  22  it  was  shown  'that  adding  a  negative  number  is 
the  same  as  subtracting  that  positive  number  whose  abso- 
lute value  is  identical.  Algebraic  subtractions  are  usually 
changed  into  algebraic  additions.  These  operations  are 
equivalent  in  results,  and  the  change  of  an  algebraic  sub- 
traction of  a  negative  number  into  an  algebraic  addition 
is  to  be  interpreted  as  illustrated  in  §  24. 

EXERCISE  XI 

Subtract  the  first  from  the  second,  and  also  the  second 
^  from  the  first  quantity  of  the  following : 

1.  2  b,  h,  7.  3  m,  4:m. 

2.  -b,2b.  '  8.  7  c,  4  c. 

3.  —a,  —2a,  9.  x-y,  —3x^y. 

4.  —  a,  2  a.  10.  7  a%,  —  8  a^b, 

5.  a,  2  a.  11.  —  ahj,  —  3  a^y. 

6.  a,  -2  a.  12.  5  0(?y\  - 13  af/. 


26  y  '.EMENTARY  ALGEBRA  [Ch.  II,  §  36 

&.  ,  CTION  OF   POLYNOMIALS 

36.  Let  A  =  b  +  c  —  d  +  e,  and  F=  m  —  n  +p  —  q.  The 
subtraction  of  the  second  from  the  first  polynomial  is 
indicated  thus: 

A  -  F=  (b  +  c  -  d  +  e)  -  (m  -  n  +  p  -  q).  (1) 

Or,  A  =  (b  +  c-d-^  e) 

F=(m  —  n-\-p  —  q^ 
A  —  F=  b  +  c  —  d  +  e  —  m  +  n  —p  +  y.  (2) 

The  quantity  A  —  F  must  evidently  be  added  to  F  to 
produce  A ;  and  the  quantity  b-\-c— d-\-e  —  m-\-n  —p  +  q 
must  evidently  be  added  to  m^n-\-p  —  q  to  make 
b-\-  c  —  d  +  e.  Expressions  (1)  and  (2)  are  identical; 
hence,  to  subtract  a  polynomial  from  a  second  polynomial : 
Write  the  first  polynomial  after  the  second^  changing  all  the 
signs  of  the  terms  of  the  first  polynomial ;  combine  like  terms. 

1.   Subtract  2a^-5ab-3b^  from  a'-2ab  +  b\ 
(a^ ^2 ab  +  b')  -  (2 a' -  5 ab-Sb") 

^d"  -^2  ab  +  b^  --2  a^  +  5  ab  +  3b% 

Or,  a2-2a&+     b^ 

2a^-5ab-Sb^ 

The  number  —  a^  must  evidently  be  added  to  2  a^  to  make  a^ ; 
3  a6  to  —5ab  to  make  —  2  a& ;  4  6^  to  —Sb^  to  make  &-. 

The  work  can  be  still  further  abridged  by  the  method  of 
Detached  CoefiBicients. 

a2-2a&+    52 

2     -5      -3 


Ch.  II,  §  36]  ADDITION   AND   SUBTRAC  ^JION  21 

2,    Subtract  m^  —  2  m  + 1  from  3  m^  -^  j^, 

37)1^ -7 m  +  1       ^v^  ' 

1  -2      +1 

2  m^  —  5  m 

(3  m^-  7  m  + 1)  -  (m^-  2  m  + 1)  =  2  m^-  5  m. 
In  each  of  the  above  expressions  take  m  =  1 ;  then 
(3^7  +  l)-(l-2  +  l)=2-5, 
or  (-3)-(0)  =-3, 

-3  =  ~3. 
The  results  check,  and  the  subtraction  is  therefore  correct. 

EXERCISE  XII 

Subtract  the  first  from  the  second,  and  also  the  second 
from  the  first  expression  of  the  following : 

1.  x  +  5^  x  +  S.  11.  4:x^  i/  +  5x. 

2.  rz:— 5,  x  —  3.  12.    a  —  5,  5. 

3.  x  +  5,  x-S.  13.   7,  2  a +  5. 

^.  X  —  5,  x  +  8.  14.  —  a;,   —  y  —  3. 

^.  5  +  x,  3  +  x.  15.  a  —  b,  b  +  a. 

6.  5  — .T,  3  —  x.  16.  3  —  :^,  n  +  1. 

7.  b  +  x,  S--X.  17.  a  — 8,  b  — 8. 

8.  5  —  x,  3  +  x.  IB.  4:  —  n,  n  +  4. 

9.  a,  a  +  1.  19.  _^ 4-8 6,   —b  —  le. 
10.  a,  a —  bo  20.  a +  b  —  c,  2a +  b  —  c. 

21.  5^  +  25  +  6,  7a-b-8. 

22.  4^2-752  +  7,  4a2  +  752__i^ 

23.  Gm  —  Bn—p,  —m  —  Sn—p  —  q. 

24.  ^3k  +  m  —  5n  +  4p,  Qk--m  +  6n  +  7p 


28  ELEMENTAIlf  ALGEBRA        [Ch.  II,  §§  37-39 

AGGREGATIONS 

37.  An  aggregation  symbol  preceded  by  the  plus  sign  may 
he  neglected,  because  the  expression  within  the  aggregation 
symbol  is  to  be  added  to  the  preceding  number,  which 
number  is  sometimes  0. 

a+\h  +  Si-=a+'b  +  C]  0  +  (a  —  6  +  c)  =  a— Z)  +  c. 

An  aggregation  symbol  preceded  by  the  minus  sign  can  be 
removed  by  changing  tJie  sign  of  every  term  contained  within 
it;  because  the  indicated  process  of  subtraction  is  the 
addition  of  the  several  terms  changed  in  sign  but  having 
the  same  absolute  value  by  §  24, 

Thus, 

7a-(a-2&--3c)=7a-a  +  26  +  3c  =  6a  +  26+3c. 

38.  By  §  37,  the  terms  of  a  polynomial  can  be  enclosed  by 
a  symbol  of  aggregation  ivhich  is  preceded  by  the  plus  sign 
without  change  of  sign  ;  and  can  be  enclosed  by  a  symbol  of 
aggregation  preceded  by  the  minus  sign  if  the  sign  of  every 
term  be  changed. 

-  3  i»2  H-  4  o;^  -  2/2  =  -  (3  a;2  ^  4  a;?/  +  2/^. 

39.  An  aggregation  enveloping  several  aggregations 
can  be  removed  by  the  foregoing  principles.  Either  the 
inner  or  the  outer  symbol  may  be  removed  first. 

Thus,  simplify  a  —  [a  —  J2  a  —  (3  a  —  6) }]. 

a-[a-{2a~(3a-6)n=^^~[«-12a-3a+&.n, 

=  (X—  [a  —  2a  +  3a—  &], 
=  a— a  +  2a— 3a  +  6| 


Ch.  it,  §  30]        ADDITION  AND   SUBTRACTION  29 

or,  removing  first  the  outer  symbol, 

a-  la-  [2  a  -  (3  a  -b)l^  =  a  -  a+  \2  a-  (S  a-b)], 

=  a  —  a  +  2  a  —  (3  a  —  &), 
=  a  —  a  +  2a--3a  +  6j 
=  -  a  +  6. 

EXERCISE  XIII 

Simplify  the  following  expressions : 

2.  (^a-b)  +  c-l(d  +  e)-f-(g-h}]. 

3.  a-lb-{c-d)]-le  +  {f-g-)-h}. 

4.  a-[6-(c  +  c?)  +  e]-(/-^)  +  7t. 

5.  [(a  -  J)  +  (c  -  c^)]  -  [(e  +/)  +  (g-  70]. 

6.  [(a  +  5)-(.  +  (?)]  +  [(e-/)-(^  +  A)]. 

7.  [(a_5)-(c-(?)]-[(.-/)-(^-A)]. 

8.  [(a  +  J)  +  (c-cZ)]  +  [(e-/)-(^-A)]. 

9.  (3a^  +  52/)-[(7a;-22/)-(8aj-4y)]  +  (2:-^). 
10.  (7  m  -  4)  +  3^-  [(8^  +  3j9  -  2)  +  5 m -  (8^ - js)] 

12.  a-[2a-(^3a-7a|-3c)]. 

13.  a-[-(-{-3a-(2a-J)|)]. 

14.  m  — [— 71— [  — 3«  — (4m  — 6w)|]. 

15.  a  -  [  ^ 6  -  (c  +  <Z)  J  +  I  e  +  (/  -  ^  +  ^)  -  (^  +  Z  ~  m)  j 
—  (w  — t))]. 

16.  a-(5 +{c  +  2a;|- Jy-z|). 

^17.    ^-(2a;-3^-[3a;- 2«/-(4a;-3y)]). 


CHAPTER   in 

MULTIPLICATION   AND   DIVISION 

MULTIPLICATION 

40.  In  §  25  there  was  given  a  definition  of  the  product 
of  two  algebraic  numbers ;  the  rule  for  finding  the  prod- 
uct ;  and  a  statement  of  the  Law  of  Signs. 

41.  The  product  of  three  algebraic  quantities  is  the 
product  of  the  first  two  quantities  multiplied  by  the  third. 

Thus,  a  '  h  '  c  =  {ah)  •  c. 

The  product  of  four  algebraic  quantities  is  the  product 
of  the  first  three  quantities  multiplied  by  the  fourth  ;  and 
so  on. 

Thus,  a  •  &  •  c  •  c?  =  {ah)  >  c  *  d^.  (ahc)  •  d. 

The  absolute  value  of  the  product  of  three  or  more  alge- 
hraic  quantities  is  the  product  of  their  absolute  values^ 
and  is  positive  when  it  contains  an  even  number  of  negative 
factors^  and  negative  when  it  contains  an  odd  number  of 
negative  factors. 

Thus,  the  product  of  —  a,  h,  —  c,  and  d  is  abed ;  and  the 
product  of  —  a,  —  h,  —  c,  and  d  is  —  ahcd. 

Since  0,  §  23,  =^a-a,  a(a- a)=  a^- a2=  0  ;  a  •  0  =  0, 

42.  By  definition,  §  7,  a^  =  aaa,,  and  a^  =  aa. 

Therefore,  a^  x  a^=  aaa  y^aa^  aaaaa  =  a^  =  a"^^. 

30 


Oh.  Ill,  §§43,44]     MULTIPLICATION  AND   DIVISION  31 

^4  X  a^  =  aaaa  x  aaaaa  =  aaaaaaaaa  =  a^  =  a^"^^. 

In  the  above  examples  the  exponents  are  positive  whole 
numbers   or   integers.     Restricting,  for  the  present,  ex- 
ponents  to   positive   integers,    the   product   of   any   two 
powers  of  the  same  letter  may  be  found  thus: 
a"*  =  aaa  taken  to  m  factors, 
a""  —  aa  taken  to  7i  factors, 
therefore, 

a^  *  a^=  (a  taken  to  m  factors)  x  (a  taken  to  n  factors), 
=  a  taken  to  (m  +  n)  factors, 
=  a"*+^. 
In  the  same  way,  a""  -  a^  -  a*  =•  a'^+^+'' 

The  principle  just  shown  is  called  the  Index  Law,  — the 
exponent  of  the  product  of  ttvo  powers  of  the  same  letter  is 
the  sum  of  the  exponents  of  the  factors. 

43.  The  process  of  multiplication  is  subject  to  three - 
fundamental  laws  (whose  truth  is  assumed),  of  which  the 
first  is  the  Commutative  Law,  —  the  product  of  two  or  more 
quantities  is  independent  of  the  order  of  the  factors. 

Thus,  2  •  3  =  3  •  2 ;  and,  in  general,  a  •b  :=b  »  a. 

44.  Multiplication  is  also  subject  to  the  Associative 
Law, —  the  product  of  three  or  more  quantities  is  independent 
of  the  order  in  which  the  factors  are  grouped  in  finding  the 
partial  products. 

Thus,  by  §  41,  5  .  4  .  3  =  (5  .  4)  .  3  =  20  .  3  =  60, 

and,  by  §  41,  5  •  4  •  3  =  5  (4  •  3)  =  5  •  12  =  GO, 

and,  in  general,      a  *  h  -  c  =  (a  -  h)  '  c  =  a  (h  »  c). 


32  ELEMENTARY    ALGEBRA  [Cii.  Ill,  §  45 

MULTIPLICATION   OF   MONOMIALS 

45.    1.  Find  the  product  of  2  ahx^y  and  5  a%x^. 
By  the  associative  law, 

(2abx^y)  (5a^bx^)  =2  -  a  -  b  -  x^  -  y  -  5  -  a^  -  b  -  a^, 

by  the  commutative  law,     =  2  *  5  '  a  '  a^  -  b  *  b  -  xr  -  x^  •  y, 

hj  the  associative  law,        =  10  aVx^y. 

2.    Find  the  product  of  —Sx^^   —  5  x^y^^  and  4  xi/z. 

By  the  associative  law, 

(_3a)2)  (_5ajy)  (4.xyz)=:{-3)'x\-~  5) -a^ -y'^  (A)  -  x  -  y  -  z, 
by  the  commutative  law,  =  (—3)  (—5)  (4-)  >  x^  -  x^  -  x  »  y^  *  y  -  z, 
by  the  associative  law  and  law  of  signs, 

=  60  a^yh. 

Hence,  the  product  of  several  monomials  is  found   5y 

annexing  to  the  product  of  the  numerical  factors  each  literal 
factor^  giving  to  it  an  exponent  luhich  is  the  sum  of  the 
exponents  of  this  factor  in  the  monomials, 

EXERCISE   XIV 

Perform  the  multiplications  indicated  : 
lo    3  a;^/  •  —  X7/^.  6.    2a^  '  4ax  '  —  11  a'^x^. 

2.  -—  a  •  —  a^  •  —  h^,  7.    —  772%"^  •  —  m%  •  —  7  mn^. 

3.  ah'Sac.-5hc,  8.    ia^  -  Gxf- -IBaWrY- 

4.  2  mn  •  —  3  m*^  •  —  4  n^,        ****  9.    a^  •  —  5  b^x  •  —  3  a^x"^, 

5.  2c  '  4:xy  -7  ab.  10.    11  ac  - —lilc^  - —13  a^c^. 

11.  ah  -  —  ac  '  —  be  -  —  cd  '  —  abed, 

12.  2  a7 .  -  3  a^  .  -  3  a^  .  -  3  a2  .  -  2  ^. 


Ch.  Ill,  §§  46-48]     MULTIPLICATION  AND   DIVISION  33 

MULTIPLICATION   OF   A  POLYNOMIAL   BY   A   MONOMIAL 

46.  An  entire  expression  is  an  expression  no  term  of 
which  contains  a  literal  quantity  in  its  denominator. 

Thus,  |a^  +  2  ab  +  b^  is  entire. 

A  fractional  expression  is  an  expression  in  which   at 
least  one  term  has  a  literal  quantity  in  the  denominator. 

3 

Thus,  - — ;:  +  2  ab  +  b^  is  fractional. 

47.  The  degree  of  a  monomial  is  found  by  taking  the 
sum  of  the  exponents  of  the  literal  factors. 

Thus,  3  a^b^G  is  of  the  sixth  degree ;  and  13  x  is  of  the  first 
degree. 

The  degree  of  a  polynomial  is  found  by  taking  the  sum 
of  the  exponents  in  that  term  in  which  the  sum  is  greatest. 

Thus,  a^  —  oaV  +  d^e'^  is  of  the  eighth  degree  because  the 
sum  of  the  exponents  of  ab^  is  eight. 

A  homogeneous  expression  is  one  in  which  the  degree 
of  the  several  terms  is  identical. 

Thus,  a*  +  4  a"&  +  6  a^b'^  +  4  aZ>^  +  b^  is  a  homogeneous  expres- 
sion of  the  fourth  degree. 

48.  The  definition  of  the  product  of  two  numbers,  §  25, 
applies  to  two  expressions  in  the  form  3(2  +  4). 

By  definition,  §  25, 

3(2+4)=3  +  3  +  etc.  to  (2  +  4)  terms, 

by  associative  law,  §  29,      =  (3  +  3  +  etc.  to  2  terms)  +  (3  + 

3+ etc.  to  4  terms), 

by  definition,  §  25,  =3.2  +  3.4, 

similarly,  a(b  +  c)  =  ab -]- ac. 

Note  :  The  above  law  is  assumed  to  hold  for  positive  fractions 
and  negative  numbers. 


34  ELEMENTARY   ALGEBRA  [Ch.  Ill,  §  48 

By  the  commutative  law,  a  (b -\-c)  =  (b  +  c)  a, 
by  the  commutative  law,  ab-\-ac  =  ba  +  ca, 

therefore,  a  (b  +  c)  =  (b  -\-  c)  a  =ab  +  ac  =  ba  -\- ca. 

The  statement  of  the  foregoing  principle  is  the  third  law 
of  multiplication,  the  Distributive  Law,  —  the  product  of  a 
(^entire')  polynomial  hy  a  monomial  is  found  by  multiplying 
each  term  of  the  polynomial  by  the  monomial  and  adding  the 
products  thus  obtained. 

1.    Find  the  product  of  2x^  —  bxy  —  2  y"^  by  3  x. 

S  X  (2  x" -5  xy -2  y'')  =  (2  x'' -5  xy -2  y^)  .  ^  X, 
=  6  a;^  —15  x^y  —  6  xy^. 


The  work 

may 

also  be 

arranged  thus : 

f^ 

V--   5xy  - 

-2y' 

\Sx 

1 

-15x'y- 

-6xy^ 

EXERCISE 

XV 

Perform  the  indicated  multiplications : 

1.  (?(2a  +  J).  6.   xyQx  +  y). 

2.  p(3m  — 4^).  7.   a(^a—2b  +  Sc}. 

3.  SxQx  —  ly^.  8.   kmn{ik  —  8m  —  7n'). 

4.  5a(a-6).  9.    6  (2  a  +  5b -9  c). 

5.  Sn^ip-q).  10.    (-l)(-5a  +  6  6-(?). 

11.  (a-lb  +  c^i^n). 

12.  (lla-8b-5c)(-Sy). 

13.  C-^ab-Sbc  +  icdX-^^d)' 


Ch.  Ill,  §49]      MULTIPLICATION  AND   DIVISION  35 

MULTIPLICATION  OF  A  POLYNOMIAL  BY  A  POLYNOMIAL 

49.    The  product  of  two  polynomials  is  expressed  thus, 
(a  +  6)(6'  +  cZ). 
By  definition,  §  25, 

(a  +  6)  (c  +  d)  =  (c  +  d)  +  (c  +  d)  +  etc.  to  (a  +  h)  terms, 
by  associative  law,  §  29, 

=  [(c  +  d)  -{-  (c  +  d)  +  etc.  to  a  terms]  + 
[(c  +  d)  +  (c  +  d)  +  etc.  to  b  terms], 
by  definition,  §  25,      z=  (c -{- d)  a  +  (c  +  d)b, 
by  distributive  and  by  commutative  laws, 
=  ac  +  ad  +  bc  +  bd. 

From  the  foregoing  principle  is  derived  the  following 
Rule  for  the  Product  of  Any  Polynomials:  Multiply  each 
term  of  the  multiplicaiid  hy  each  term  of  the  multiplier  and 
add  the  successive  products. 

1.    Find  the  product  of  2x^  —  ?»xy  +  4:  y'^  and  x  —  y. 
Arrange  the  work  thus : 

X  -       y 


2  0^  —  3  xhj  4-  4  xy^ 

—  2  x^y  +  3  xy^  —  ^y^ 


2  X?  —  ^  x^y  +  7  xy^  —  4  ?/^ 

^  The  product  Qii2x^  —  Zxy  -\-4  y'^  and  x  is  written  in  the  third 
line,  and  the  product  of  2  aj^  —  3  a?^/  +  4  y^  and  —  2/  in  the  fourth 
line.  Like  terms  are  arranged  in  columns  so  that  they  may  be 
united. 

The  product  of  three  or  more  polynomials  is  found  by 
taking  the  product  of  the  first  two  by  the  third,  and  so  on. 


36  ELEMENTARY  ALGEBRA       [Ch.  lit,  §§  50,  51 

50.  A  polynomial  is  said  to  be  arranged  with  reference 
to  a  letter  when  the  powers  of  that  letter  constantly  in- 
crease or  decrease.  Any  letter  can  be  selected  as  the 
letter  of  order.  If  the  exponents  of  the  letter  increase, 
the  polynomial  is  said  to  be  arranged  in  ascending  order. 

Thus,  x^-\-3  x]f-  —  3  a?y  —  y%  arranged  with  reference  to  x, 
in  descending  order,  is  x^  —  3  iJi^y -{- 3  xy^  —  y^ ;  and  the  same 
expression,  arranged  with  reference  to  y,  in  descending  order, 
is  —2/^  +  3  xy^  —  3  xhj  +  x\ 

51.*  The  application  of  the  method  of  Detached  Coeffi- 
cients will  be  facilitated  if  all  of  the  terms  of  the  expres- 
sions to  be  multiplied  are  arranged  with  reference  to  a 
single  letter  in  the  same  order,  the  coefficients  of  missing i 
powers  of  the  letter  of  arrangement  being  represented  hy\ 
zero. 

Multiply  a3  +  a%  +  aJ2  +  l^  by  a^  -  h\ 

1+1+1+1 
1  +  0-1 
1+1+1+1 

1  + 1  +  0  +  0  - 1  - 1  =  a^  +  a^6  -  a&4  -  5^ 

The  result  obtained  may  be  checked  by  substituting  a=h=i., 
(14.1  +  1  +  1)(1_1)  =  1  +  1_1_1;   4.0  =  0. 

Detached  coefficients  are  most  advantageously  employedi 
in  finding  the  products  of  homogenous  expressions. 

The  above  example  also  illustrates  the  following  prin- 
ciple :  The  product  of  two  homogeneous  expressio?is  is  a  homo^ 
geneous  expression  whose  degree  is  the  sum  of  the  degrees  of  the 
multiplicand  and  multiplier. 


\ 

Vn,  III,  §  51]       MULTIPLICATION  AND   DIVISION  37 

EXERCISE  XVI 

Perform  the  indicated  multiplications : 

1.  a  +  2  by  a  +  3.  8.  -x  +  2  hj  x—7. 

2.  a  — 3  by  a  — 4.  9.  a  —  4t  hy  2a +  1. 

3.  m  +  4  by  m  —  3.  10.  2  a  +  5  by  a  —  4. 

4.  n  — 2  by  n  +  5.  11.  2  a— 7  by  3  a  +  6. 

5.  x-2  hj   —x  +  S.  12.  3a +  4  by   —3a +  5. 

6.  —  x—S  hj  x  +  4:.  13.  x^  +  xi/  +  ^^  hj  X  —  y. 

7.  —^  +  3  by   —a; +  6.        14.  x^  —  xy  +  y*^   by  x  +  y, 
15.  2a^  +  ah  +  lfi  by  2a -5. 

le.  ?^x'^-'Qxy  +  ^  hy -- x  +  ^y. 

17.  a2  +  a5  +  52  by  a2  -  a5  +  5^. 

18.  x^  +  x'^y''- +  y^  by  :r4-:r2/  +  /. 

19.  a^  +  3a26  +  3aJ2  +  J3  by  a  +  h  +  c. 

20.  a^  -  4  a^i  +  6  a2J2  -  4  ^53  _j.  54  ^y  ^  __  5^ 

21.  3a2-.4a^  +  5&2  by   2a2-3a5  +  2J2. 

22.  b  x^  —  y^ -\-H x^y  —  4  2:2/2  by   2 :i;2  +  3 2/2  —  :ry. 

23.  10a252__i3^4_6^3^_f_g^53+354  by  x^-\-f-xy. 

24.  a*-7a26-8a52-lla36  by  a2:?;-.  5  a2:2  4- 7  a^  -  2;3 

25.  m*^  —  11  aW"  —  4  :r2/^  —  3  n^  by  ^^  —  7  —  2  alfi  +  m^. 

26.  a2  —  a6  +  a;  —  ?/  by  a2  —  a6  —  ^  +  y. 

27.  :r3_2:i:2  4-:?;-3  by  2;3_2^2  +  ^_3, 

28.  2^-3rc2-:r  +  2  by  2rz;3-3:i:2  +  ^_2. 

29.  5a3-4a2J  +  2a62--63  by  2  aJ2- J3_  5  ^3  + 4  ^25. 

30.  7?  +  y'^  '\-  z^  by  a;2  +  2/2  -f  ;22  _  ^^  __  ^^  _  y^^ 


38  ELEMENTARY   ALGEBRA       [Ch.  Ill,  §§  52-54 

DIVISION 

52.  In  §  26  there  was  given  a  definition  of  division  of 
two  algebraic  numbers,  the  rule  for  finding  the  quotient, 
and  a  statement  of  the  law  of  signs. 

If  the  indicated  divisor  be  zero,  since  the  product  of  a 
finite  number  and  0  is  0,  it  follows  that  the  quotient  can- 
not be  found;  that  is,  0  cannot  be  used  as  a  divisor. 

If  the  dividend  be  zero,  since  a  •  0  =  0,  -  may  be  defined 
as  0. 


53.    Since,  by  §42, 

dr  .  a""^  aj^-^'^. 

by  §  26, 

a^                                        ' 

nnrl     "hv  S  9.(\ 

^m-\-n 

This  principle  is  called  the  Index  Law, — the  exponent 
of  the  quotient  of  two  powers  of  the  same  letter  is  the  exponent 
of  the  dividend  minus  the  exponent  of  the  divisor. 

Note,  m  and  n  are,  as  in  §  42,  positive  integers  only ;  and  m  and 
n  are  restricted  to  such  values  that  m  is  not  less  than  n,  A  full  dis- 
cussion will  be  found  in  Chapter  XVII. 

DIVISION  OF   MONOMIALS 

54.  From  §§  26,  45,  and  53,  the  quotient  of  two  mo- 
nomials is  found  hy  annexing  to  the  quotient  of  the  numerical 
factors  each  literal  factor  whose  exponent  is  its  exponent  in 
the  dividend  minus  its  exponent  in  the  divisor. 

1.    Divide  8a;3  by  2x\ 
2ar 


Ch.  Ill,  §  55] 

2.    Divide  12  h^c^m^  by 
12  6Vm3 


MULTIPLICATION  AND   DIVISION 

=  —  4  b^~V~'^m^~^  =  —  4  b'^cm. 


39 


—  3  bciri^ 

55.  Since  any  quantity  divided  by  itself  produces  1,  it 
is  evident  that  a"*  -j-  a"*  =  1 ;  and,  by  the  Index  Law,  it  is 
also  evident  that  —=  a'"'''  =  a^.     The  quotients  just  de- 

rived  must  be  equal,  because  the  dividends  and  divisors 
are  identical.  Hence,  an^/  finite  quantity/  with  the  exponent 
zero  may  he  defined  as  equal  to  1;   or,  a^  =  1. 

Divide  -  30  a'^b^c  by  -  6  a^he. 

-  30  a'b'c 


'  6  a'^bc 


:  5  a^-462-1^1-1^  5  a%c'  =  5  •  1  •  &  •  1  =  5  6. 


EXERCISE  XVII 

Perform  the  indicated  divisions : 


1, 


2^^ 


""    Z^<^ 


J 


8. 


2.    - 


3^ 


10. 


4. 


■-^^i   "• 


6. 


7. 


a" 

12  a^b 
-4  6* 

-  25  a2/)3 
5^262 


—    o  o;^?/        "7 
20  a%^      ' 


39  a3x3 

13  a3:r3 

-  28  «*x2 

-  7  d'x 

-  64  a562 

- 16  a*62 

-  30  a*6V. 

13. 


14. 


—  6  a^6y 

—  34  'jfixf'z^ 
Vlx^yz^ 

91  x^y^z"^ 
44mV> 

—  4  TO%2 


15. 


16. 


17. 


18. 


19. 


20. 


21. 


-33ffl769c"i 
11  aSJ^cS 

60  gS^^cis 

-  15  a36V2" 

84aJ9512g23 

-  7  ai253(ji6' 

63  'jfiy'^gy^ 
7  a^y  ai2  ■ 

-  78  ofiy'z^"' 
- 13  a;2?/42>*' 

42  aWc^^ 

-  7  a«^*ci5' 

-b2a^3?y'^z^ 
- 13  a^xhj^z^ 


40  ELEMENTARY   ALGEBRA  [Ch.  Ill,  §  56 

DIVISION  OF  A  POLYNOMIAL   BY  A   MONOMIAL 

56.  It  has  been  shown,  §  48,  that,  by  the  distributive  law, 
ab  -{-  ac=  a(b  +  c^.  By  the  definition  in  §  48,  if  the  prod- 
uct ah  +  ac^  and  the  factor  a  are  given,  the  quotient  will 
be  b  +  c, 

^Whence  is  derived  the  following  Rule  for  the  Division  of 
a  Polynomial  by  a  Monomial :  Divide  each  term  of  the  poly- 
nomial by  the  monomial  and  add  the  quotients  thus  derived. 

Divide  a^  -  2  a^J  +  8  a^J^  by  a^, 

-^ =  1  —  2  a&  +  8  arb^. 


EXERCISE  XVIII 

Perform  the  indicated  divisions : 

2.  .  5.  - . 

xy  —  Zx^ 

^     5  m2  +  10  m3  ^6  a^^  4.  g  aH^  +  16  ^^ 

3. .  6.     — —  —'. 

—  bm  „       . ,      2 a;* 

y     -  21  x^y  -  91  xf  +  56  y^ 
-ly 

Q     16  a;8y8  -  48  x'^y^'^  +  112  a;^ 


9. 


16  xY 
gilici  -13  aWc^  -  21  gS^Sgia 


^Q     51a:yi-102a^V^ 


Cii.  Ill,  §  57]       MULTIPLICATION  AND  DIVISION  41 

DIVISION   OF   A  POLYNOMIAL  BY  A   POLYNOMIAL 

57.  Since  it  is  always  true  in  exact  division  that  the 
product  of  the  divisor  and  quotient  gives  the  dividend, 
and  since  (oi?  —  xy  +  ^")  (x  —  y^  =  x^  —2  x^y  +  2  xy'^  —  y^, 
it  is  possible  to  take  either  x^  ~xy  +  y'^^  ov  x  —  y^  as  the 
divisor,  and  the  other  expression  as  the  quotient;  while 
^3  _  2  x^y  +  2  xy'^  —  y^  is  the  dividend.  Take  x^—  xy  +  y'^ 
as  the  divisor.     Then 

(ofi  —  2  x^y  +  2  xy*^  —  y^^  -r-  (aP'  —  xy  +  y'^^  =  x  —  y. 

The  quotient  x  —  y\^  derived  from  the  dividend  and  divisor 
by  the  following  process : 

Notice  first  that  the  dividend  and  divisor  are  both  arranged 
in  descending  powers  of  x.  The  first  term  of  the  dividend  is 
evidently  the  product  of  the  first  term  of  the  divisor  and  the 
first  term  of  the  quotient,  the  first  terms  in  each  case  being 
evidently  the  term  of  highest  degree  because  of  the  order  ox 
arrangement.  Therefore,  a?^,  the  first  term  of  the  dividend, 
divided  by  a^,  the  first  term  of  the  divisor,  gives  x,  the  first 
term  of  the  quotient. 

Now  the  first  term  of  the  quotient  is  a  multiplier  of  each 
term  of  the  divisor,  as  will  be  seen  by  referring  to  the  case, 

(pi?  —  xy  +  y^)  (x  —  y)  =  x^  —  2  x^y  +  2  xy'^  —  y^. 

Therefore  the  partial  products  of  all  the  terms  of  the  divisor 
by  the  first  term  of  the  quotient  form  a  part,  at  least,  of  the 
dividend.     That  is, 

(x^  —  xy  +  y^)x  =  x^  —  x^y  +  xy^ 

must  be  subtracted  from  the  dividend  since  the  dividend  is  the 
sum  of  the  partial  products  found  by  multiplying  all  the  terms 
of  the  divisor  by  all  the  terms  of  the  quotient.  The  remainder, 
so  derived,  is 

x^  —  2x^y  +  2xy^  —  y^^  (x^  —  x^y  +  xy^  =  ~-x^y  +  xy^  —  jf 


42  ELEMENTARY  ALGEBRA  [Cpi.  Ill,  §  57 

This  remainder  may  be  considered  as  a  new  dividend  and  is 

the  product  of  the  divisor  and  the  remaining  term  (or  terms) 

of  the  quotient. 

As  before,  the  first  term  of  the  remainder  (new  dividend)  is 

the  product  of  the  first  term  of  the  divisor  and  the  first  term  of 

the  quotient.     Hence  —  x^y  divided  by  oc^  gives  —  y,  the  second 

term  of  the  quotient.     Since  the  second  term  of  the  quotient  is 

a  multiplier  of  each  term  of  the  divisor,  the  product  of  the 

whole  of  the  divisor  and  the  second  term  of  the  quotient  is 

sought. 

{x^  —  xy  +  f'){-y)  =  -  x^y  +  xy^  —  y^. 

Subtracting  this  product  from  the  remainder,  the  new  remainder 
will  be  0 ;  that  is,  the  division  is  exact. 

The  above  explanation  may  be  expressed  thus: 

a^  ^2  Qi?y  +  2  xy'^  —  y^  =  (q(^  —  x^y  +  xy^)  +  (—  x^y  +  xy'^  —  y^), 

a?  — 2  xhj  +  2  xy'^  —  y'^  _a?  —  xhj  +  xy^      —  x^y  +  xy'^  —  y^ 
x^  —  xy  +  y^  x^  —  xy  +  y^  x^  —  xy  +  y^     ' 

-x-y. 

It  will  be  noticed  that  the  dividend   is   separated  into  such 
terms  that  each  may  be  exactly  divided  by  the  divisor. 

The  following  arrangement  is,  therefore,  more  convenient : 


ividend  =^x^  —  2x^y  -^2  xy^  —  if 

0?  —  xy  +  y^  —  Divisor 

x^  —    x^y  +    xy^ 

x  —  y  =  Quotient 

^     x^y+    xy^  —  / 

If  the  quotient  contains  more  than  two  terms,  the  process  of 
division  is  the  same. 

Checking  the  division  by  substituting  x=zy=^ly 

(qi?—  2  xhj  +  2  xy^  —  ]f)-^{^  —  xy  +  y^)  =  00'—y, 

(1^2  +  2-1) -(1-1  +  1)  =  1-1, 

0-1=0. 


Ch.  Ill,  §  58]       MULTIPLICATION  AND   DIVISION 


43 


58.  From  the  foregoing  principle  is  derived  the  fol- 
lowing Rule  for  the  Division  of  a  Polynomial  by  a  Poly- 
nomial : 

1.  Arrange  both  polynomials  in  the  descending  or  ascend- 
ing order  of  some  common  letter, 

2.  Multiply  each  term  of  the  divisor  by  the  quotient  ob- 
tamed  by  dividing  the  first  term  of  the  dividend  by  the  first 
term.  ^/  the  divisor,  "  ' 

3.  Subtract  the  partial  products  so  derived  from  the 
dividend, 

4.  With  the  remainder  still  arranged  in  the  same  order  as 
before^  coritinue  the  process  until  there  is  no  remainder^  or 
until  the  degree  of  the  first  term  of  the  divisor  is  higher 
than  that  of  the  first  term  of  the  remainder. 


1.   Divide  m^  —  3  m^n  +  3  mn^  —  n^  by  m- 


n. 


m^  —  3  m^n  +  3  mn^  ■ 

-ii' 

m 

—  n 

w?  —    ni?n 

w? 

—  2  mn  +  n^ 

—  2  m^n  4-  3  mv? 

—  2  Kri?n  +  2  mii? 

-n' 

-n' 

2.    Divide  2a^-5a%  +  l  a%'^^bab^+2¥  by  a^-ab  f  J2. 


2a^-^  a%  +  7  a'b'  -5ab^  +  2b' 

2  g^  -  2  a^6  +  2  a^b^ 

-  3  a^6  +  5  a'b^  -^5ab^  +  2b' 
-3a^b  +  Sa'b'-3ab^ 

2d'b'-2ab^-^2b^ 
2a'b^-2ab^4-2b^ 


ab-^    b^ 


2a^-3ab  +  2l/ 


3,   Divide  15 x^  -^  7 x  +  7 x^  +  15 x^  +  4:  hj  1  +  S x^ -\'  2x. 
Arrange  the  dividend  and  divisor  in  the  same  order. 


44  ELEMENTARY  ALGEBRA 

By  the  method  of  detached  coefficients : 


[Ch.  Ill,  §  58 


3  +  2  +  1 


5-1  +  4 


15+   7  +  15  +  7  +  4 
154-104,   5 

-    34,10  +  7+4 
•>-   3-   2-1 

12  +  8  +  4 

12  +  8  +  4 

The  quotient  5  —  1+4  must  have  x^  in  the  first  term,  and  an 
integer  only  in  the  last  term ;  and  is,  5  aj^  —  a?  +  4. 

by  x^~-?>x^-\-2x  +  l. 


l__l_2  +  5-4  +  l  +  l 

I4-O-3  +  2  +  I 


-1+1+3- 

_l_0  +  3- 

-5+1+1 
-2-1 

1  +  0- 
1  +  0- 

-3+2+1 
-3+2+1 

1  4.0-3+2  +  1 


1  -1+1 

x^  —  x  +  1 


The  divisor  x'^  —  3o(?-\-2x-\-l  contains  no  term  in  a?\  since^ 
times  X*  equals  0,  to  make  the  method  available,  the  a^  appears 
with  the  coefficient  0. 

In  detaching  coefficients,  the  coefficient  of  any  missing  powei 
of  the  letter  of  arrangement  is  always  written  as  0. 


EXERCISE  XIX 

Perform  the  indicated  divisions : 

1.  a^  +  5a  +  Q  hj  a  +  2. 

2.  x^  —  2x  —  ?>hyx  +  l. 

3.  x^  -  16  by  x  +  4:. 

4.  2;2  -  14  2^  +  49  by  x—  7. 


Cii.  Ill,  §  58]       MULTIPLICATION  AND   DIVK  ION  45 

5.  4  a2  _  12  a  +  9  by  2  a  -  3. 

6.  36a^-60a;  +  25  by  6a;-5. 

7.  4a2  +  12aJ  +  952  by  2«  +  3J. 

8.  9  a;^  —  6  xmn  +  rw^n^  by  3  a;  —  raw. 

9.  0^-21  by  a:- 3. 

10.  64  +  a^  by  4  +  a. 

11.  6 ma;  —  8 am  —  9a;+12a  by  Sx  —  4a. 

12.  21  ax  —  B5  ay  +  S  bx  —  5  b^  by  3  a;  —  5  y. 

13.  20  ac  -  15  a(^  -  12  6c  +  9  5(^  by  5  a  -  3  6. 

14.  a^  +  b^  +  c^  +  2ab  +  2ao  +  2be  hy  a  +  b  +  e. 

15.  p^  +  q^  +  r^  +  2pq  —  2pr—  2  gr  hy  p+q  —  r. 

16.  ^2_|_  2  jjg  +  ^2  _  ^  by  ^  +  5'  +  r. 

17.  12a2_452_5^  +  2a5  +  4ac-9Jc  by  2  «  -  5  -  (r. 

18.  1  -  18  a2  +  81  a*  by  1  -  6  a  +  9  a\ 

19.  -  2  a;li..'La:L4-S2.a;2  +  145  a;  +  72  by  9  +  8  a;  -  a;2, 
2a)  216  a^  +  125  by  36  a^  _  30  «  +  25. 

21.  1  -  S2p^  by  1  +  2jt»  +  4 j92  +  Sp^  + 16 j9*. 

22.  128  a*^-3  -  160  a562  +  2  cjSJ  +  15  a^  by  3  a2  _  8  a5. 

23.  5  «c  +  75c  +  3  a2  _  7  a5  -  6  52  -  2  c2  by  a  -  3  6  +  2  c. 

24.  44^-30-16^/2  + 3:r  +  9a^  by  4?/  +  3x-5. 

25.  48  a;2- 192  a;?/ +  192/ -27  22  by  4a;-8j/  +  33. 

26.  4  a;*  -  197  a;2^2  +  49  ^4  by  2  2^2  +  15  a;y  +  7  3/2, 

27.  80  6c  +  18  a  -  64  62  _  48  6  -t-  9  a2  +  30  c  -  25  c2  by  3  a 
-  8  6  +  5  c. 

28.  a'^-Qac  +  9<fi-ib^-  4:bd-cP  by  a  +  2b-Se-i-d. 

29.  a^  +  63  +  c3  -  3  a6c  by  a  -+  6  +  c?. 

30.  x^  —  1/^  hy  X  —  y. 


46  ELEMENTARY   ALGEBRA  [Ch.  Ill,  §  58 

EXERCISE  XX 

Perform  the  indicated  operations  and  check  the  results  : 

1.  a  [a  +  (6  —  c)]  -^  (a  +  6  —  c). 

2.  5a;-(3a;-4)-[7x(2-9a;)]. 

3.  (a2  +  ah  +  P) ia-b)  -  3(a3  -  53). 

4.  7a-2[3a-25+f(a  +  5)-(a-J)J], 

5.  [-c  +  (a  +  J)][-c-(«  +  6)]-(6'2-a2)-(c  +  a)j,^ 

6.  (a-6)(a  +  J-c)  +  (6-c)(6  +  c-a) 

+  (c  —  d)(c  +  a  —  6). 

7.  (a  + J)(a  — 5)  —  ^(a4-5  — c)  —  (5  — a  — c) 

+  (6  +  e  — a)na-5  — c}. 

8.  ix^  -  3/2)  ^  (a;  +  3/)  +  2.l{x  -  y)(a;2  -  2  a:y  +  «/2)] 

9.  8  a  H- 4  a  +  7  -  [6  a  H- 2]  .  3  -  6  a2  H- 2  «  -  7(1  -  a). 

10.  (a;2— ?/2  — ^2^  2  yg) -i- (a;— ?/  +  3). 

11.  ^a-\h-a-i(2a+h-\a-h\')']. 

12.  a-2[2a-6-(3a-25-f4a-3J|)]. 

13.  (cc2_2a;+l)(a^-3a;2+3a;-l). 

14.  (a^- 32)^  (a;* +.2  2:3 +4  2^24.  8  a; +16). 

15.  (15x*+7a;3  +  15a:2+7a.+  4)^(3a^4.2a;+l). 

16.  (t"— y^} -i-Qx'^Jr  afiy  +  x^y'^+xy^  + y'^').  . 

17.  (l  +  2jo  +  4j92+8j93  +  16p*)(l-2p). 

18.  (21f)a3+i25)H-(36a2_30a+25). 

19.  (a^  +  3/2  -  22)  (a;2  +  t/2  +  ^2)  _  (a;2  _  2/2  +  22)  (a;2  _  ^2  _  2-2) 

20.  (2a;2-3a;3/  +  3/2-3)(3-2/2_3a;y  +  2a;2). 

21.  (77a963-55a8i2_35a85  +  25a')-5-(5a-7a26). 


CHAPTER  IV 

EQUATIONS   AND  PROBLEMS 

59.  An  algebraic  equa^tion  is  a  statement  that  the  nu- 
merical values  of  two  expressions  are  the  same.  The  ex- 
pression preceding  the  sign  of  equality  is  called  the  first 
(or  the  left)  member,  and  the  expression  following  the  equal 
sign  is  called  the  second  (or  the  right)  member. 

Thus,  in  the  equations,  a  +  b  =b  +  a,  and  a;  + 1  =  3,  the  first 
members  are  respectively  a  +  b,  and  a?  + 1 ;  and  the  second 
members  are  respectively  b  +  a,  and  3. 

60.  If  an  equation  is  always  true  for  any  values  of  the 
letters  involved,  it  is  called  an  equation  of  identity.  In 
the  more  advanced  work  the  sign  of  higher  equality  in 
identities  is  written  = . 

Thus,  a  +  b  =  b  +  aj  for  all  values  of  a  and  b. 

61.  Equations  other  than  identical  are  called  conditional 
equations,  since  the  equality  does  not  hold  for  all  values 
of  the  letters  involved,  but  is  conditional  upon  a  certain 
value. 

Thus,  a;  -f- 1  =  3,  if  0?  =  2,  but  not  if  x  has  any  other  value. 

Equations  of  identity  are  more  briefly  called  identities ; 
and  equations  of  condition  are  more  simply  called  equa- 
tions. 

47 


48  ELEMENTARY   ALGEBRA       [Cii.  IV,  §^  g2-64 

62.  The  process  of  finding  for  the  letters  involved  those 
values  which  make  the  members  equal  is  called  solving 
the  equation.  These  values  are  called  the  roots  of  the 
equation.  The  equation  is  said  to  be  satisfied  if  the 
numerical  values  of  the  members,  found  by  substituting 
the  values  of  the  roots,  are  the  same. 

Thus,  2  is  a  root  of  the  equation  a?  + 1  ==  3,  and  the  equation 
is  satisfied  by  substituting  2  for  x]  2  +  1  =  3, 

63.  An  Axiom  may  be  defined  as  a  self-evident  truth. 
The  following  principles  are  stated  as  algebraic  axioms : 

1.  If  equal  quantities  are  added  to  equal  quantities^  their 
sums  will  he  equal, 

2.  If  equal  quantities  are  subtracted  from  equal  quanti- 
ties,, the  remainders  will  he  equal. 

3.  If  equal  quantities  are  multiplied  hy  the  same  quan- 
tity or  hy  equal  quantities^  the  products  will  he  equal, 

4.  If  equal  quantities  are  divided  hy  the  same  quantity  or 
hy  equal  quantities^  the  quotients  will  he  equal, 

5.  Quantities  equal  to  the  same  quantity^  or  equal  quanti- 
ties^ are  equal  to  each  other. 

The  axioms  should  be  memorized  in  order. 

64.  In  an  algebraic  problem  the  quantities  whose  values 
are  given  are  called  known  quantities,  and  are  usually 
represented  by  the  first  letters  of  tlie  alphabet;  those 
quantities  whose  values  are  not  given  but  are  to  be  deter- 
mined are  called  the  unknown  quantities,  and  are  usually 
represented  by  the  last  letters  of  the  alphabet. 

Thus,  in  a? -1-3 =11,  a;  is  the  unknown  quantity,  and  3  and  11 
are  the  known  quantities ;  in  y  -\-b  =  3  a,  y  is  the  unknown 
quantity,  and  b  and  3  a  are  the  known  quantities. 


C^ii.  IV,  §§  65,  66]    EQUATIONS   AND   PROBLEMS  49 

65.  A  simple  equation  is  one  which  in  its  simplest  form 
contains  only  the  first  power  of  the  unknown  quantity. 

Thus,  Sx  +  T  =  28,  and  3x  =  6  a^  are  simple  equations. 

The  solution  of  a  simple  equation  may  depend  upon 
any,  or  all,  of  the  axioms. 

1.  Solve  for  x^  the  equation  2x  +  S  =  11. 
Since  2  cc  +  3  =  11, 

byAx.  1,  2a;  +  3-3  =  ll-3, 

by  §23,  2x  =  S, 

by  Ax.  4,  x  =  4:. 

The  root  may  be  tested  and  the  equation  satisfied  by 
substituting  the  value  of  the  root  in  the  given  equation. 

Thus,  2(4)4-3  =  11, 

11  =  11. 

66.  The  process  of  satisfying  an  equation  is  variously 
called  verification,  testing,  and  checking.  The  root  should 
always  be  substituted  in  the  given  equation. 

2.  Solve  for  y,  the  equation  22/  +  7  =  y  —  4. 


Since ' 

2y  +  7  =  y-4, 

by  Ax.  1, 

■   2y  +  7  -l  =  y-A-7, 

by  §  23, 

22/  =  2/-4-7, 

combining, 

2y  =  y-ll, 

by  Ax.  1, 

2y-y  =  y-y-ll, 

by  §23, 

2y-y=-ll, 

combining, 

2/= -11. 

Vekifioation  :  2  (— 11)  -f  7  =  — 11  —  4, 

- 15  =  -  15. 


50  ELEMENTARY   ALGEBRA       [Ch.  IV,  §§  67, 68 

67.  By  the  use  of  Axiom  1,  a  term  may  be  changed 
from  the  first  to  the  second  member,  or  vice  versa. 

If  0^  +  4  =  5, 

by  Ax.  1,  ic  +  4  —  4  =  5  —  4, 

or,  -'  i»  =  5  — 4. 

The  +  4  in  the  first  member  appears  in  the  second  member 
as  —4. 

Again,  if  2  a;  —  7  =  a?  +  10, 

by  Ax.  1,  2x-7  +  7  =  x  +  10+7, 

or,  2x  =  x  +  17y 

and,  by  Ax.  1,  2  x—x  =  x  —  x-{- 17, 

or,  X  =  17. 

Vekific  ATiox :        2  (17)  —  7  =  17  + 10, 

34__7  =  27, 

27  =  27. 

The  process  of  changing  a  term  from  the  first  to  the 
second  member,  or  vice  versa^  is  called  transposition ;  ani/ 
term  may  he  transposed  if  its  sign  he  changed. 

Transposing  in  the  equation  6a;  — 11 +  3  =  2a;  +  l  —  ic, 

5aj-2a;  +  a;=ll-3  +  l. 

It  is  to  be  noticed  that  transposition  is  simply  an  appli- 
cation of  Axiom  1. 

68.  From  the  foregoing  principles  is  derived  the  follow- 
ing Rule  for  the  Solution  of  a  Simple  Equation :  Transpose 
all  the  unhnoivn  terms  to  the  first  memher  and  all  the  known 
terms  to  the  second   memher;  comhine  similar  terms;  and 

\   divide  both  members  hy  the  coefficient  of  the  unknown  quantity. ' 


Cii.IV,  §68]  EQUATIONS   AND  PROBLEMS  61 

1.  Solve  for  a:,  the  equation 

6:?;-22  =  4a:-30.  (1) 

Transposing  in  (1),        6  a;  -  4  a;  =  22  -  30,  (2) 

uniting  in  (2),  2x  =  -S,  (3) 

applying  Ax.  4  in  (3),  a?  =  —  4.  (4) 

Verification  :        6 (-  4)  -  22  =  4 (-  4)  -  30, 
-24-22  =  -16-30, 
-46  =  -46. 

2.  Solve  for  x^  the  equation 

3(:^-7)  +  3  =4-2(6+2:).  (1) 

Simplifying  in  (1),  3  a;  -  21  +  3  =  4  - 12  -  2  a;,  (2) 

transposing  in  (2),  3  a;  +  2  a;  =  21  -  3  +  4  - 12,  (3) 

uniting  in  (3),  5x  =  10,  (4) 

applying  Ax.  4  in  (4),  x=2. 

Verification  :        3  (2  -  7)  +  3  =  4  -  2  (6  +  2), 
^15  +  3  =  4-16, 
.       -.12  =  -12. 

3.  Solve  for  x^  the  equation 

(2  a:  +  3)  (3  a;  +  1)  =  (6  2;  +  1)  (:i:  +  5)  -  22.  (1) 

Simplifying  in  (1), 

6  x^ +  11  x  +  3  ==6  x" +  31  x  + 5-- 22,       (2) 
transposing  in  (2), 

6x'-6x''  +  llx-Slx  =  -S  +  5-~22,  (3) 

'uniting  in  '(3),  ~  20  a;  =  -  20,  (4) 

applying  Ax.  4  in  (4),  x  =  l. 

Verification  :      (2  +  3)  (3  + 1)  =  (6  -4- 1)  (1  +  6)  -  22, 

20  =  42-22, 
20  =  20. 


52  ELEMENTARY  ALGEBRA  [Ch.  IV,  §  68 

EXERCISE  XXI 

Solve  for  x,  the   following   equations  and   verify  the 
results : 

1.  7x-5  +  22;  =  13. 

2.  4  +  12  x  =  a;  +  15. 

3.  34a;=6:r:  +  5  +  51. 

4.  70-3a;-2a;  =  7a;-2. 

5.  x=Q  x  +  l —  bx  —  lQ. 

6.  a;  +  2  a;  +  3  a; +  4  a;  =  100. 

7.  -2a;-5=  9a;  +  5a;  +  21-68-6. 

8.  0  =  5  a;  +  7  a;  -  9  a;  -  11 X  +  107  a;  -  74  +  a;  -  26. 

9.  3 a; -(a; -7)  =  a; +  15. 

10.  3 a; -(2a; -8)  =  19. 

11.  3(a;  +  l)-2a;  =  93. 

12.  81  -  4  (a; +  1)  =  a; +  7. 

13.  103-3(a;-5)=2x  +  18.  . 

14.  13(2x-l)=5(5a;  +  4>. 

15.  25-6(a;-6)  =  20-(2a;-13). 

16.  2(9-a;)  +  5(2a;  +  3)  =  81. 

17.  6(20+3a;-l)-5(8a;-7)  +  19  =  2(a;-72). 

18.  3  •  5  (x  +  6)  +  5  •  7  (1  +  2  a;)  -  7  •  9  (a;  -  8)  =  827. 

19.  (2a;-l)(3a;  +  l)=(6a;-12)(a;+3). 

20.  (5  X  +  7)(6  a;  -  3)  =  (10  a;  +  2)(3  X  +  2)  -  9. 

21.  7(a;-l)-3(l-a;)  =  -4(6  +  a;). 

.22.  3(2a;  +  7)+4(6  +  a;)=-4(a;-3)  +  3(2a;  +  l)-10. 

23.  6(2  x-  4)  -  3(2  a;-  1)  =  7(3 a;  +  2)  -  8(4 a;-  2). 


Ch.IV,  §69]  EQUATIONS   AND   PROBLEMS  53 

69.    If  the  equation  contains  fractions,  it  can  be  simpli- 
fied by  application  of  Axiom  3. 

1.    Solve  for  x^  the  equation 

^-4  =  10-a;.  (1) 

b 

Applying  Ax.  3  in  (1),         a;  -  24  =  60  -  6  a?,  (2) 

transposing  and  uniting  in  (2),     1  x  =  84,  (3) 

applying  Ax.  4  in  (3),  x  =  12.  (4) 

12 
Verification  :  -- —  4  =  10  —  12, 

6 

2  -  4  =  10  - 12, 

_2  =  -2. 


2.    Solve  for  x^  the  equation 


8       4:  —  X         X 


9       11 
Simplifying  in  (1), 


'lQx  +  2      x  +  2' 
33  9 


\(4:-x)^x      16x  +  2     x-{-2 
99  3  33  9/ 


(1) 


(2) 


applying  Ax.  3  in  (2), 

•    S(A-x)  =  33x-3(16x  +  2)+il(x  +  2),    ^(3) 
simplifying  in  (3), 

32-8a;  =  33a;-48a;-6  +  lla;  +  22^  (4) 

.  .f      .  V 

transposing  and  uniting  in  (4), 

-4a?=-16,  \  (6) 

applying  Ax.  4  in  (3),  07  =  4.  (ft) 

^r  84-44      64 +  2,  6 

VEKincATiON :  9  •  -JT  =  3  -  -3f-  +  9^ 

0  =  |-2  +  |. 


54  ELEMENTARY  ALGEBRA  [Ch.  IV,  §  69 

EXERCISE  XXII 

Solve  for  x,  the  following  equations : 

6.  ^  +  ?-6  =  l. 

8      6 

7.  -  -1. 

„     8(2  +  5a;)-5_92;  +  2 
9  2 

7  4.   19-(7+|)  =  |  +  7.         9.    |-|  +  |-£=18. 
5.    5x-^=lx-l-  10.    8a;-?  =  ^  +  153. 


XT 

Q                   a;                  X    +    ^ 

y±. 

^      9~     3     ■ 

1/2. 

5  +  |  =  .-5. 

-'S. 

^-5  =  a:-23. 
4 

10 


2 

7a;      1      17      H/o      ,  in 
"^-    T-4-T8  =  36^^"  +  '^- 

2(7.r-l)_3(3rr  +  5). 

7a;+13     a;  +  8     a^  +  ll 
13. = • 

16  13  8 

14.  30(a;-2)  +  f  =  ^iJ+30. 

15.  1(5  a;  + 1)  -  K4  a^  +  5)  =  i(3  a:  -  1)  -  2^(6  a;  +  4), 
Sx  +  9  ,  5  a;  -  33      48  -  x  ,  x-\l  _'6  +  x 


16. 


72  36  9  4  24 


X     x-2     a;-22     a;-12^32-a; 
■    8        5  10  20  40    " 

^3_    3^-5 _ 4(2^+4)  _r9-. ^-71     ^_  15^ 
16  9  L    2     ^    12  J 

20.    (4;»-l)C5»;  +  })  =  (2»  +  J)(10a;-J). 


Ch.IV,  §70]  EQUATIONS   AND   PROBLEMS  65 

70.  The  statement  of  a  problem  in  algebraic  language 
often  leads  to  an  equation.  The  problem  is  solved  by 
finding  the  numerical  value  of  the  numbers  which  first 
appear  as  unknowns.  Certain  relations  of  the  unknowns 
in  definite  numbers  are  given  ;  from  these  relations  the 
values  of  the  unknowns  are  determined. 

Little  difficulty  need  be  met  in  translating  the  state- 
ment of  a  problem  into  algebraic  language  if  it  be  remem- 
bered that  every  algebraic  expression  represents  some 
number. 

EXERCISE  XXIII 

1.  What  is  the  value  in  cents  of  2  two-dollar  bills, 
3  dollar  bills,  4  quarters,  and  5  nickels  ?  oi  a  two-dollar 
bills,  b  dollar  bills,  c  quarters,  and  x  nickels  ? 

2.  If  X  is  the  tens'  digit  and  4  the  units'  digit  of  a 
number  of  two  digits,  what  is  the  number  ? 

3.  If  3  is  the  tens'  digit  and  x  the  units'  digit  of  a 
number  of  two  digits,  what  is  the  number  formed  by 
reversing  the  order  of  the  digits  ? 

4.  If  in  a  number  of  three  digits  the  tens'  digit  is  x^ 
and  the  hundreds'  digit  is  twice  the  tens'  digit,  and  the 
units'  digit  is  four  times  the  tens'  digit,  what  is  the 
number  ? 

5.  What  is  the  cost  of  20  articles  bought  at  the  rate 
of  3  for  X  cents? 

6.  If  x  represents  a  certain  digit,  what  is  the  next 
higher  digit  ?     the  next  lower  digit  ? 

7.  If  a;  is  a  certain  digit,  what  are  the  2  next  higher 
(consecutive)  digits? 


66  ELEMENTAKY  ALGEBRA  [Ch.  IV,  §  70 

8.  If  X  is  an  odd  number,  what  are  the  next  two  even 
numbers  ?  the  next  two  odd  numbers  ? 

9.  If  X  men  contribute  equally  to  a  certain  fund  of 
%  225,  how  much  does  each  man  contribute  ? 

10.  If  a  man  spends  a  dollar  a  day  more  than  on  the 
preceding  day,  and  on  the  tenth  day  spends  x  dollars, 
how  much  does  he  spend  on  the  twenty-third  day  ? 

11.  If  the  price  of  eggs  is  lowered  3  cents  a  dozen 
from  the  original  price  of  a  cents  a  dozen,  how  much  does 
one  Q^g  now  cost  ? 

12.  If  the  interest  on  a  certain  sum  of  money  for  a 
given  time  is  computed  at  x  per  cent,  what  will  be  one  per 
cent  higher  rate  ? 

13.  What  is  the  value  in  cents  of  the  same  number, 
Xy  of  dollars,  cents,  quarters,  and  dimes  ? 

14.  If  in  a  certain  number  of  two  digits  the  units'  digit 
is  x^  and  the  tens'  digit  is  four  times  the  units'  digit,  what 
is  the  sum  of  the  digits  ? 

15.  If  a  newspaper  increased  x  per  cent  over  the  pre- 
ceding yearly  circulation  at  the  end  of  each  year,  and 
if  the  circulation  at  the  end  of  the  first  year  was  25,000, 
what  was  the  circulation  at  the  end  of  the  second  year  ? 

16.  If  the  rate  of  a  stream  is  2  miles  per  hour,  what 
will  be  the  rate  down  the  river  of  a  crew  which  rows  4 
miles  an  hour  in  still  water  ?  up  the  river  ? 

17.  What  is  the  perimeter  of  a  rectangular  field  whose 
length  is  a  feet  and  whose  breadth  is  h  feet  ? 

18.  What  is  the  greater  of  two  numbers  if  the  greater 
is  three  times  the  excess  of  the  less  number,  x^  over  12  ? 


Ch.IV,  §71]  EQUATIONS   AND   PROBLEMS  67 

71.  After  the  conditions  of  a  problem  have  been  stated 
in  algebraic  language,  the  next  step  is  to  find  two  equal 
expressions.  In  th.e  equation  formed  of  these  two  equal 
expressions  the  roots  are  found  by  §  68. 

1.  The  sum  of  a  number  and  its  double  is  48.  Find 
the  number. 

Let  X  =  the  number, 

then  2  0?  =  double  the  number, 

and  x-\-2x  =  ^x  =  the  sum  of  the  number  and  its  double, 
but  48  =  the  sum  of  the  number  and  its  double, 

by  Ax.  5^         3  0?  =  48, 
by  Ax.  4,  0?  =  16. 

Verification:       16  +  2  (16)  ==  48, 

48  =  48. 

2.  Find  that  number  which  lacks  as  much  of  18  as  it 
exceeds  10. 

Let  X  =  the  number, 

then  18  —  ic  =  the  amount  the  number  lacks  of  18, 

and  a?  — 10  =  the  amount  the  number  exceeds  10, 

I  but  the  amount  the  number  lacks  of  18  is  the  same  amount 
that  the  number  exceeds  10  ; 

by  Ax.  5,  18-a;  =  r»-10, 

or  —  2  0?  =  —  28, 

by  Ax  4,  X  =  14. 

Verification  :  18  — 14  =  14  — 10, 

4  =  4. 


58 


ELEMENTARY  ALGEBRA 


[Ch.  IV,  §  71 


3.    A's  age  exceeds  B's  by  25  years.     Five  years  ago  A 
was  six  times  as  old  as  B.     Find  the  age  of  each. 


Let 
then 
and 
and 
and 
but 

by  Ax.  6, 
simplifying, 
uniting, 
by  Ax.  4, 

Verification 


X  =  B's  age, 
25  +  a?  =  A's  age, 
a;  —  5  =  B's  age  5  years  ago, 
25  +  oj  —  5  =  A's  age  5  years  ago, 
^{x  —  5)  =  ^  times  B's  age  5  years  ago, 

20  +  ^  =  A's  age  5  years  ago, 
Q(x-S)  =  20  +  x, 
6a;~30  =  20  +  a;, 
5a;  =  50, 
a;  =  10. 
6(10-5)  =20  +  10, 
30  =  30. 


4.    The  units'  digit  of  a  number  is  double  the  tens' 
digit,  and  the  sum  of  the  digits  is  12.     Find  the  number. 

Let  a?  =  tens'  digit, 

then  2x  =  units'  digit, 

and  x  +  2x  =  sum  of  the  digits, 

but  12  =  sum  of  the  digits^ 

by  Ax.  5,  x  +  2  x  =  12, 

uniting,  3  a;  =  12, 

by  Ax.  4,  a;  =  4, 

by  Ax.  3,  2  a?  =  8, 

Therefore  the  number     =  10  (a;)  +  2  a;  =  48. 

Verification  :        4  +  8  =  12, 
12  =  12 


Ch.IV,  §71]  EQUATIONS   AND   PROBLEMS  59 

5.    The  sum  of  the  third  part  and  twelfth  part  of  a 
number  is  25.     Find  the  number. 


Let 

x  = 

=  the 

number, 

then 

X  _ 
3" 

=  the  third  part  of  the  number, 

and 

X 

12" 

=  the  twelfth  part  of  the  number. 

and 

X        X 

3"^  12" 

=  the 

sum  of  the  third  and  twelfth  parts. 

but 

25  = 

=  the 

sum  of  the  third  and  twelfth  parts, 

by  Ax.  5, 

3  +  12-^^' 

by  Ax.  3, 

4:X  +  x  =  300, 

uniting, 

5  a?  =  300, 

by  Ax.  4, 

x  =  m. 

Verification  : 

60  ,60     ^^ 
¥  +  12  =  ^^' 

20  +  5  =  25, 

25  =  25. 

6.  A  man  has  the  same  number  of  half-dollars,  quarters, 
dimes,  and  nickels.  Find  the  number  if  he  has  all  together 
13.60. 

Let  X  =  the  number  of  each  coin, 

then  50  a;  =  the  value  of  the  half-dollars  in  cents, 

and  25  x  =  the  value  of  the  quarters  in  cents, 

and  10  a;  =  the  value  of  the  dimes  in  cents, 

and  6  a;  =  the  value  of  the  nickels  in  cents, 

and  90  a;  =  the  values  of  all  the  coins  in  cents, 

but  360  =  the  values  of  all  the  coins  in  cents, 


60  ELEMENTARY   ALGEBRA  [Cii.  IV,  §  71 

by  Ax.  5,  90i»  =  360, 

by  Ax.  4,  a;  =  4. 

Verification  :  90  (4)  =  360, 

360  =  360. 

EXERCISE  XXIV 

1.  The  sum  of  a  number  and  three  times  that  number 
is  48.     What  is  the  number? 

2.  The  sum  of  10  and  twice  a  number  equals  four 
times  that  number.      What  is  the  number? 

3.  If  13  be  subtracted  from  eight  times  a  number,  the 
remainder  equals  86.     What  is  the  number  ? 

4.  If  five  times  a  certain  number  is  subtracted  from  27, 
the  remainder  is  7.     Find  the  number. 

5.  Five  times  a  number  exceeds  twice  that  number  by 
21.     Find  the  number. 

6.  Find  that  number  th>e  sum  of  whose  products  by  S 
and  4  respectively  equals  119. 

7.  One  number  is  twice  another  number  and  theiu 
difference  is  14.     Find  the  numbers. 

8.  The  sum  of  12  and  three  times  a  number  equals 
the  excess  of  39  over  six  times  the  number.  Find  th( 
number. 

9.  Twice  a  number  lacks  as  much  of  20  as  three  timer 
the  number  exceeds  20.     Find  the  number. 

10.    Twelve  times  a  number  exceeds  7  as  much  as  tei 
times  the  number  lacks  of  15.     Find  the  number. 


Ch.IV,  §71]  EQUATIONS  AND  PROBLEMS  61 

11.  The  sum  of  12  and  four  times  a  number  exceeds  by 
2  nine  times  the  number.     Find  the  number. 

12.  The  excess  of  four  times  a  number  over  24  equals 
the  sum  of  9  and  the  number.     Find  the  number. 

13.  The  greater  part  of  8  equals  three  times  the  smaller 
part.     Find  the  parts. 

14.  Three  times  the  smaller  part  of  15  exceeds  by  5 
twice  the  larger  part.     Find  the  parts. 

,  15.    The  sum  of  two  numbers  is  47,  and  their  difference 
is  3.     Find  the  numbers. 

16.  The  sum  of  two  numbers  is  26,  and  their  difference 
is  6.     Find  the  numbers. 

17.  The  sum  of  two  numbers  is  120,^  and  the  greater 
exceeds  the  less  by  21.     Find  the  numbers. 

18.  The  difference  of  two  numbers  is  26  and  their  sum 
is  52.     Find  the  numbers. 

19.  The  excess  of  7  over  the  larger  part  of  5  equals 
twice  the  smaller  part.     Find  the  smaller  part. 

20.  The  sum  of  three  consecutive  numbers  is  39.     Find 
the  numbers. 

21.  Find  the  ages  of  A  and  B  if  the  sum  of  their  ages 
is  62  years,  A  being  16  years  older  than  B. 

22.  A  has  four  times  as  much  money  as  B,  and  both 
have  $  125.     How  much  has  each  ? 

23.  A,  B,  and  C  have  together  I  28.     A  and  B  each  has 
three  times  as  much  as  C.     How  much  has  each  ? 


62  ELEMENTARY  ALGEBRA  [Ch.  IV,  §  71 

24.  A  has  twice  as  much  money  as  B,  and  B  has  three 
times  as  much  as  C.  All  have  together  $150.  How 
much  has  each^? 

25.  A  has  twice  as  many  dollars  as  B,  three  times  as 
many  as  C,  and  half  as  many  as  D.  If  they  all  have  $92, 
how  much  has  each? 

26.  A,  B,  and  0  together  have  $  54.  If  A  has  twice  as 
much  as  B,  and  G  has  as  much  as  A  and  B  together,  how 
much  has  each? 

27.  A  and  B  together  have  $  12 ;  B  and  C,  $  15 ;  A  and 
C,  $  19.     How  much  has  each  ? 

28.  The  same  number  each  of  dollars,  dimes,  and  cents 
amount  to  $8.88.     Find  the  number  of  cents. 

29.  The  sum  of  a  certain  number  of  quarters  and  four 
times  that  number  of  cents  is  $  5.80.  Find  the  number  of 
cents. 

30.  A  has  ten  times  as  many  cents  as  dimes  and  eight! 
times  as  many  dimes  as  dollars.  If  he  has  in  all  $  13,  findi 
the  number  of  dimes.  M 

31.  A's  age  exceeds  B's  by  20  years.  Ten  years  ago  A 
was  twice  as  old  as  B.     Find  the  age  of  each. 

32.  A  is  now  four  times  as  old  as  B  ;  5  years  ago  he 
was  seven  times  as  old  as  B.     Find  the  age  of  each. 

33.  A  is  now  five  times  as  old  as  B  ;  in  12  years  he  will 
be  three  times  as  old  as  B.     Find  the  age  of  each. 

34.  Six  years  ago  a  father  was  six  times  as  old  as  his 
son,  whose  age  now  lacks  30  years  of  the  fathers  age. 
Find  the  age  of  each. 


Ch.  IV,  §  71]  EQUATIONS   AND   PROBLEMS  63 

35.  If  A  is  now  52  years  old  and  B  is  now  12,  find  the 
number  of  years  ago  that  A  was  five  times  as  old  as  B. 

36.  The  units'  digit  of  a  number  of  two  digits  is  three 
times  the  tens'  digit,  and  the  sum  of  the  digits  is  12. 
Find  the  number. 

37.  The  tens'  digit  of  a  number  of  two  digits  exceeds 
by  4  the  units'  digit,  and  the  sum  of  the  digits  is  8.  Find 
the  number. 

38.  The  tens'  digit  of  a  certain  number  of  two  digits  is 
3  times  the  units'  digit.  If  18  be  subtracted  from  the 
number,  the  order  of  the  digits  will  be  reversed.  Find 
the  number. 

39.  The  hundreds'  digit  of  a  number  of  three  digits  is 
twice  the  tens'  digit  and  four  times  the  units'  digit.  If 
297  be  subtracted  from  the  number,  the  order  of  digits 
will  be  reversed. '    Find  the  number. 

40.  A  fifth  of  a  certain  number  exceeds  the  eighth  of 
that  number  by  6.     Find  the  number. 

41.  The  excess  of  a  certain  number  over  8  equals  a 
third  of  that  number.     Find  the  number. 

42.  The  quotient  of  a  certain  number  divided  by  9 
exceeds  the  twelfth  part  of  the  number  by  1,  Find  the 
number. 

43.  The  twelfth  part  of  a  certain  number  is  8  less  than 
the  sixth  part  of  that  number.     Find  the  number. 

44.  The  eighth  part  of  a  certain  number  is  3  less  than 
the  fifth  part  of  that  number.     Find  the  number. 

45.  The  ninth  part  of  a  certain  number  exceeds  by  1 
the  tenth  part  of  that  number.     Find  the  number. 


64  ELEMENTARY   ALGEBRA  [Ch.  IV,  §  71 

46.  The  third  part  of  a  certain  number  exceeds  5  by 
as  much  as  the  eighth  part  is  less  than  6.  Find  the 
number. 

47.  The  fifth  part  of  a  certain  number  exceeds  7  by  as 
much  as  the  ninth  part  is  less  than  7.     Find  the  number. 

48.  Two-thirds  of  a  certain  number  exceeds  one-sixth 
of  that  number  by  15.     Find  the  number. 

49.  Three-eighths  of  a  certain  number  exceeds  one- 
fourth  of  that  number  by  4.     Find  the  number. 

50.  Two-thirds  of  a  certain  number  exceeds  four- 
sevenths  of  that  number  by  2.     Find  the  number. 

51.  The  sum  of  one-third  and  one-thirteenth  parts  of 
a  certain  number  is  16.     Find  the  number. 

52.  One  and  one-half  times  a  certain  number  exceeds 
three-eighths  of  that  number  by  36.     Find  the  number. 

53.  The  sum  of  the  ages  of  a  father  and  son  is  48  years. 
How  many  years  ago  was  the  son's  age  one-seventh  of  the 
father's  age  if  the  son's  age  is  now  1 2  years  ? 

54.  A  has  four  times  as  many  cents  as  dimes  and  twice 
as  many  dimes  as  dollars.  If  he  has  in  all  $5.12,  find 
the  number  of  dollars. 

55.  Find  that  number  of  three  digits  in  which  the 
hundreds'  digit  is  double  the  tens'  digit,  and  in  which 
the  tens'  digit  is  double  the  units'  digit,  if  the  sum  of  the 
digits  is  14. 


CHAPTER  V 

TYPE  FORMS   IN   MULTIPLICATION 

72.  The  products  of  certain  expressions  are  so  often 
required  that  it  is  convenient  to  have  a  shorthand  method 
of  writing  the  product  without  performing  the  multipli- 
cations as  in  §  49.  These  expressions  and  their  products 
are  called  type  forms. 

CASE  I 

73.  By  multiplication,  {a  +hy==a^  +  2ab  +  Ifi, 

Here  a  and  h  represent  the  sum  of  any  two  quantities ;  the 
square  of  the  sum  is  required. 

The  process  may  be  represented  thus : 

(1st  number  +  2d  number)2=  (1st  number)^  +  2  (1st  num- 
ber)(2d  number)  +  (2d  number)^ 

Rule  :  The  square  of  the  sum  of  two  quantities  is  the 
square  of  the  first  quantity^  plus  twice  the  product  of  the 
first  and  second  quantity^  plus  the  square  of  the  second 
quantity, 

EXERCISE  XXV 

Write  the  indicated  squares  by  inspection : 


1. 

(m  +  ^)^. 

5. 

(c  +  2  dy. 

9. 

(a  +  S2)2. 

2. 

(a  +  2  J)2. 

6. 

(2  (?  +  3  dy. 

10. 

(2  ^  +  ^)2. 

3. 

ic  +  dy. 

7. 

(6-2  +  dy. 

11. 

(4^4-  my. 

4. 

(2  c  +  dy. 

8. 

(a2  +  5)2. 

12. 

(2  a +  12(^2)2^ 

65 


1 


66  ELEMENTARY   ALGEBRA         [Cii.  V,  §§  74,  75 

CASE   II 

74.  By  multiplication,  (a  —  by  =  a^  —  2  ah  +  P. 

Here  a  —  b  represents  the  difference  of  any  two  quantities . 
the  square  of  the  difference  is  required. 

Rule  :  The  square  of  the  difference  of  two  quantities 
is  the  square  of  the  first  quantity^  minus  twice  the  prod- 
uct of  the  first  and  second  quantity^  plus  the  square  of  the 
second  quantity, 

EXERCISE  XXVI 

Write  the  indicated  squares  by  inspection  : 

1.  (m  -  ny.  5.    (7  -  5)2.  9.    (5  m-  n)2. 

2.  {n  -  my.  6.    (m-2  dy.         10.    (11  m  -  l)^. 

3.  (c-dy.         7.  {c^sdy.       11.  (1-10^)2. 

4.    (d  -  cy.  8.    (3  tZ  -  cy.  12.    (2  m  -  3  d^^y. 

CASE   III 

75.  By  multiplication,  (a  +  &)(a  —  J)  =  a^  —  P, 

Here  the  product  of  the  sum  and  difference  of  the  same  two 
quantities  is  required. 

Rule  :  The  product  of  the  sum  and  difference  of  the' 
same  two  quantities  is  the  difference  of  the  squares  of  the' 
first  and  second  quantities. 

exercise  XXVII 

Write  the  indicated  products  by  inspection  :  -v 

1.  (a  4-  c}(a  -  c).      d  ^  C       4.    (2  c  +  d)(2  c  -  dy^ 

2.  (m-n)(m  +  n).   yv^-^     5.    (a^  -\- b) (a'^  -  b) ,    \ 

3.  (^d  +  e)(id  ^  e}.  6.    (2c-d?)(2c+d:^^. 


Cii.  V,  §  76]      TYPE   FORMS   IN   MULTIPLICATION 


67 


76.  It  is  sometimes  possible  to  arrange  the  terms  in 
both  multiplicand  and  multiplier  to  take  the  form  of 
Case  III. 

1.  (a  +  b  +  c)(a  +  b-c)=l(a  +  b)  +  c]l(a  +  b)-cl 

=  (a  +  by-c^ 

=  a'  +  2ab-\-b"-c'', 

2.  (a-b  +  c)(a  +  b-c)  =  la-(b-c)\ja  +  (b-c)l 

=  a?-{b.-cf, 


'  +  2bc- 


■c^ 


The  rule  of  Case  III  applies  to  the  product  of  terms  so 
arranged. 

EXERCISE   XXVIII 


u 


Write  the  indicated  products  by  inspection : 

2.  (^ni  —  n  +  p} {ni  +  n—  p^ 

3.  (m  —  n—p^(m  —  n+p) 

4.  (^m  —n  —p^(7n  +  n  +p^ 

5.  {2a  +  b  +  c^)(2a  +  b-e^} 


(2  a2  +  3  a5  +  52)(_  2  a^  +  Sab  +  52). 

(^2  _  2  ^i  +  d^)  (6-2  +  2  ^^  -  6^2), 

(c2  -ab  +  5d)(c^  +  ab  +  5  d). 
(^s  —  sa  —  sb')(s  -{-  sa  +  sF), 
(6-2  _  s^a  +  s25)(s2  +  s^a  -  s%). 

(xY  -Sxf  +  i  2/0(^y  +  Sxf-4: 2/*). 
L'lS.    (So?  -  2x9j  +  f  -  iX^x^  -  2xy  -^  2/^  +  4). 
14.    (a^  -  3  a  J  +  52  -  2  6)  (  -  a3  -  52  _  2  5  -  3  ab). 


6. 

7. 

8. 

9. 
10. 
11. 
12. 


68  ELEMENTARY  ALGEBRA  [Cii.  V,  §  77 

CASE   IV 
77.   By  Case  I,  p, 

=  (a-^by-\-2(a  +  h)c  +  c% 
=  a'  +  2ab  +  b^  +  2aG-{-2bG  +  c^, 
=  a^  +  6-  +  c^  +  2  ab  +  2  ac  +  2  be. 
By  Case  11, 

(a-6-c)2=f(a-5)-cp, 

=  (a-6)2_2(a-5)c  +  c2, 

=  a^-2ab  +  b^-2aG  +  2bc  +  (^, 

=  a^  +  b'-{-c'-'2ab-2ac  +  2bc. 

Rule  :    The  square  of  any  'polynomial  is  the  sum  of  the 
squares  of  the  several  terms^  and  twice  the  product  of  every  i 
term  by  every  term  that  folloivs  it^  giving  to  every  product 
the  proper  sign, 

EXERCISE  XXIX 

Write  the  indicated  squares  by  inspection  : 

1.  ^a  +  h-cy,  8.  (1  + 2a +  3^2)2. 

2.  l-a  +  h  +  cf.  9.  (2a  +  5-3c)2. 

3.  Q^a-h  +  cy.  10.  (2a^-{-^hx  +  x^y, 

4.  (^a^h-.cy.  11.  (1+2 2: +  3:^:2)2. 

5.  (_^_J_^)2.  12.  (2  a2  -  3  a?>  -  5 />2)2. 

6.  (2a  +  J  +  ^)2.  13.  [(a4-fi)4-^  +  2  6?]2. 

7.  (^_(.  26 +  7^)2.  14.  (2 77^2 -  3 7^2  +  4 mn)2, 

15.  [(2a-5)-(?+36?]2. 

16.  \Jom  —  lbmn  +  n(n  —  m-yy. 

17.  (2  ^3  -  a25  +  3  a62  _  J3-)2. 


Ch.  V,  §  78]      TYPE   FORMS   IN  MULTIPLICATION  69 

CASE   V 
78.    By  multiplication, 

and  {a  -  J)3  =  a^  -  3  a%  +  3  aS^  _  J3. 

Rule  :  The  cube  of  the  sum  of  two  quantities  is  the  sum 
of  the  cubes  of  the  quantities  plus  three  times  the  product  of 
the  square  of  the  first  quantity  and  the  secondJ  plus  three 
times  the  product  of  the  first  quantity  and  the  square  of  the 
second. 

Rule  :  The  cube  of  the  difference  of  ttvo  quantities  is  the 
difference  of  the  cubes  of  the  quantities  minus  three  times  the 
product  of  the  square  of  the  first  quantity  and  the  second^ 
plus  three  times  the  product  of  the  first  quajitity  and  the 
square  of  the  second. 

The  result  of  the  two  rules  can  be  shown  thus : 
(a  ±  6)3  =  aS  ±  3  a%  +  3  aJ^  ±  J3^ 

where  the  sign  ±,  read  "plus  or  minus,"means  that  in  the 
cube  of  a  4-  J  the  signs  are  all  plus ;  and  that  in  the  cube 
of  a  —  J  the  signs  are  alternately  plus  and  minus. 
Write  by  inspection  (a  —  2  6)^. 

(a-2  6)3=  (a)3-3(a)2(2  &)  +3(a)  (2  6)^-  (2  h)\ 

EXERCISE  XXX 

Write  the  indicated  cubes  by  inspection : 

1.  (x  +  yy.  4.   (2  a +  6)3.  7.   (w-5n)3. 

2.  (x-yy.  5.    (2^:2  +  3  2/2)8.  8.    {la^-l ab'^)K 

3.  (x^J^y'^y.  6.    (2x^-''&xyz)\        9.    (1-52^2)3. 


70  ELEMENTARY   ALGEBRA  LCh.  V,  §  79 

CASE   VI 

79.  By  multiplication  the  product  of  two  binomials  of 
the  form  x  +  a  and  x  +  b  can  be  determined. 

(:r  +  2)  (a;  +  3)  =  a;2  +  5  a:  +  6. 

(^x  +  2)(x-S)  =  x^-    x-6. 

(ix-'2)(x  +  n}  =  x'^+    x-6. 

(a;-2)(a;-3)  =  2;2-5a:  +  6. 

Rule:  The  product  of  any  two  hinomials  whose  first  terms 
are  identical  is  the  product  of  the  first  terms  of  the  hinomials^ 
the  algebraic  sum  of  the  second  terms  as  the  coefficient  of  the 
common  term^  and  the  product  of  the  second  terms  of  the 
hinomials, 

EXERCISE  XXXI 

Write  the  indicated  products  by  inspection : 

1.  (x  +  V)(x  +  2^.  8.  (:ry-3)(rr2/  +  4). 

2.  (a;  +  l)(^-2).  9.  (a:2_3)(^2  +  4). 

3.  (m  +  5)(m-4)-  10.  i^  —  l  xy^(^- xy^. 

4.  (^_3)(^__4).  11.  (^2_4)(^^2+6). 

5.  (m-7)(m  +  3).  12.  (ax  +  lV)(iax  +  V), 

6.  (x-bn)(x  +  ?^n^.  13.  (a2-21)(a2  +  3). 

7.  (x^^b^(x^-S).  14.   (xy-l)(xy  +  ^y 

15.  (16-5a;^)(16-2:r^^). 

16.  (5  m^n  —  3  ny'^^  (5  rrfin  —  ny"^^. 

17.  [(a +  6) +  5]  [(a +  6) -3]. 

18.  [l-(:r  +  2/)][l-4(a;  +  2/)]. 

19.  [(a;-2/)  +  2][(a;-2/)f  7]. 


:h.  V   §  80]      TYPE   FORMS   IN   MULTIPLICATION  71 

CASE   VII 

80.  By  multiplication  the  product  of  two  binomials 
irhich  contain  the  same  letters  can  be  determined. 

(x-2y)(2x+?>y)  =  2x^-  xy  +  Qy"^. 
(x  +  2y)(2x-Zy)=-2x^+  xy-Qy^. 
(x-2y)(2x^?>y^=^2x'^-lxy  +  Qy'^. 

Rule  :  The  product  of  tivo  binomials  which  contain  the 
ame  letters  is  the  product  of  the  first  terms  of  the  binomials^ 
he  algebraic  sum  of  the  cross  products^  and  the  product  of 
he  second  terms  of  the  binomials. 

Write  by  inspection  (3  a;  +  7  ?/)(2  a:  —  4  y). 

{3  x  +  1  y){2  x-^  4:y)  =  ^  x"  +  14.xy  -12  xy  --2S  y\ 

The  cross  products,  as  14  xy  and  — 12  xy,  are  usually  com- 
ined,  without  writing  in  full,  into  the  middle  term  of  the 
•roduct. 

EXERCISE  XXXII 

Write  the  indicated  products  by  inspection  : 

1.  (2x-a)(?>x  +  a^.  7.  (x-b  y)(2x -3  y^. 

2.  (2m  +  a)(m-2a),  8.  (2  ^— m2)(3a;- m^). 

3.  (2x  +  aX?^x-a).  9.  (:r  +  1)(3  a:-4). 

4.  (2m~a)(m-2a).  10.  (5  a- 2  5)(2  5  +  5  a). 

5.  (2x—a^(Zx-a^.  11.  (a  — 11  (?)(2  a  — <?). 

6.  (2  71 -«)(n -2  a).  12.  (l-xy^i?^- b  xy^. 

13.  (6a2-7a:)(2a2  +  :r). 

14.  (Zx^  +  4:xy^{^x^-lxy^. 


72  ELEMENTARY  ALGEBRA  [Ch.  V,  §  8i: 

CASE   VIII 

81.    By  multiplication, 

(a  +  J)(a2  --ah  +  h^)=a^  +  b% 

and  (a  -  6)  (a^  +  ab  +  h^)  =a^-  bK 

Here  a  +  b  represents  the  sum  of  any  two  quantitiei 
and  a  —  b  the  difference  of  any  two  quantities. 

Rule  :  The  product  of  the  sum  of  two  quantities  and  thi 
sum  of  the  squares  of  the  quantities  minus  the  product  oj 
the  quantities  is  the  sum  of  the  cubes  of  the  quantities. 

Rule  :  The  product  of  the  difference  of  two  quantitie 
and  the  sum  of  the  squares  of  the  quantities  plus  the  prod^ 
uct  of  the  quantities  is  the  difference  of  the  cubes  of  th 
quantities, 

EXERCISE  XXXIII 

Find  the  indicated  products  by  inspection: 

1.  (x  -  y^)(x^  +  xy  +  2/2).         3.    (2  +  a)(4  -  2  a  +  a^).. 

2.  (x  +  y^(x^-xy  +  y'^).         4.    (a  -  2)(a2  +  2  a  +  4).. 

5.'   (:r2  +  4)(2;^-4:zj2_^16). 

6.  (5a-2  6)(25a2_,.10a6  +  4  62). 

7.  (2&-5a)(4J2  +  i0a6  +  25a2). 

8.  (7c-l)(43e2  +  7  6?  +  l). 

9.  (1  +  10J)(1-10  5  +  100  62). 

10.  (8  c?-a;5)(64(^2  + 8(^2:5  +  ^^^). 

11.  [a  + 6 +  c][(a  +  5)2-(a  + 5)^  +  ^2], 

12.  [3(a-e)  +  4(J-cZ)] 

[9(a  -  c)2  -  12(a  _  c) (J  _  (^)  +  16(5  -  dfl 

13.  [2(a;-2/)  +  3][4(a;-2/)2-6(^-y)  +  9]. 


Cn.  V,  §  81]      TYPE  FORMS  IN  MULTIPLICATION  ■   t3 

REVIEW  EXERCISE  XXXIV 

Write  the  indicated  products  by  inspection : 
1.    (2  a +  6)2.  4.    (a  +  7)(a-7). 

2:    (a;-ll)(a;-6).  5.  ■  (a;- 3)(a;2  + 3  a;  + 9). 

3.    (1  -  2  a;  -  3  «/)2.  6.    (l-4:xy. 

7.  (2cc  +  2/  +  l)(2a;  +  y-l). 

8.  (3ic2-5  2/2)2. 

9.  (3  a;2  +  4  x/) (3  a:2  _  4  2:^2). 

10.  (2to  — 3«)(3m  — 4«). 

11.  (a;  +  4)(a^-4a;  +  16). 

12.  (2  ^2  -  w2  -  J92) (2  m2  -  n2  +  p2). 

13.  (2  W2  -  n2  _  j,2-) (-2  ^2  _  „2  _  j,2^. 

14.  (2a;2-5«/2)(5a;2_2/). 

15.  (14-a)(3-a). 

16.  (1  — 4art/3)3. 

17.  (2  :J?^2  _  22)2, 

18.  C8x-a')(^9a^  +  Sxa  +  a^'), 

19.  (7  a2  +  3  icy)  (7  a2  +  3  a;«/). 

20.  (5  a;y  —  3  «6) (5  a;^/  +  3  a5). 

21.  (5  xi/  + 1)  (25  a;2y2  _  5  3;,^  +  1). 

22.  (3m  — 2n  +  y)(3m  +  2«  — ^). 

23.  (a2  +  aa;  +  a^)  (a2  —  aa;  +  a^). 

24.  (a  +  a;4-2/)(a  — a;  — y). 

25.  (aa;  + m  — n)(a2;  — <^). 

26.  (m  —  n  — a:  + ?/)(m  — n +  a;  — ?/). 

27.  (2  aa;  -  3  bt/)  (4  a22;2  +  6  abxjj  +  9  %2-). 

28.  (l  +  x  +  a^^Cl  —  x  +  a^). 


74  ELEMENTAKY  ALGEBRA  [Ch.  V,  §  81 

29.  (l+X  +  X^y^l  +  x  +  X^).  1 

30.  (l+X  +  X^)(l  +  X-x'^'). 

31.  [(m  +  2w)  +  (a  +  5)]2. 

32.  (7mV-4£?)2. 

33.  (2a2  +  7  5)(2a2_7  5). 

34.  [(2a  +  36)  +  (3c  +  (;)]a.  J 

35.  j2(a-5)-c|2. 

36.  (1  -  5  a;  +  2  2;2)2. 

37.  (2a2_3aJ  +  4)(_2a2-3a5+4). 

38.  i^x^y^-lxyz^y'. 

39.  (7  a;2|/22  —  4  xyz^')^. 

40.  |2(a-Z.)-3(c  +  cZ)|2. 

41.  (2^  -  t/2  +  3)  (3?'  -  /  -  11). 

42.  1 3(a  +  5)  +  6  w2(a2  +  W')  \\ 

43.  (2w  — 4n)(3m  +  Ji). 

44.  (a  +  5  — c2)(a  +  6  +  c2). 

45.  (5  M?/  —  cz) (1  ny  —  A  cz). 

46.  (a2_3  52)(a4  +  3a252  +  9  54-), 

47.  f(2a  +  5)  +  3c2||(2a+5)-3c2j. 

48.  [x— ?/+ 3a][a;— ?/— 2  a]. 

49.  [2(a-5)  +  3^][2(a-5)-2c]. 

50.  (26'  +  2cZ  +  5a  +  5  5)(2c  +  2cZ-5a-55). 

51.  [3(a  +  5)-2c][2(a  +  6)  +  5c]. 

52.  (a;2  —  xy  —  22)3 

53.  [a-3(2:-2/)][2a-4(x-y)]. 

54.  sl3(a  +  5)-5(a2+52)|^13(a4.5)  +  5(a24.52)^, 

55.  [7(?n-w)-(a-6)][3(TO-m)  +  4(a-6)]. 


CHAPTER  VI 

FACTORING 

82.  If  a,  J,  and  ^  are  limited  to  integral  expressions  and 
if  a  -  h  ==  c^  then  a  and  b  are  called  factors  of  (?,  and  c  is 
called  a  multiple  of  a  and  of  b. 

An  integral  expression  is  prime  when  it  has  no  factors 
except  itself  and  1. 

The  process  of  finding  the  prime  factors  of  an  integral 
expression  is  called  factoring. 

83.  The  factors  of  a  monomial  can  be  obtained  by  in- 
spection. 

Thus,  the  factors  of  36  aV  are  2«2«3-3-a-a«a«5«6. 

Tlie  factors  of  a  polynomial  are  indicated  by  the  form 
of  the  expression,  which  is  often  one  of  the  type  forms  dis- 
cussed in  the  previous  chapter. 

CASE  I 

84.  When  each  of  the  terms  of  the  expression  contains  a 
common  monomial  factor. 

1.  Factor  a^  +  S  x^  +  5  x. 

By  inspection  x  is  common  to  each  term. 
Therefore,  a^  +  Sa:^  +  5x==  x(p?  +  3  a?  -4-  S), 

2.  Factor  ba^  +  oab  +  10a. 

By  inspection  5  a  is  a  factor  of  each  term. 

Therefore,     5  a^  +  5  a6  + 10  ce  =  5  a(a  +  &  4-  2) 

75 


76  ELEMENTARY  ALGEBRA  [Ch.  VI,  §  84 

EXERCISE  XXXV 

Factor : 

1.  x^  +  ax.  6.  —  12  ax*^  +  4  axy  —  8  ay\ 

2.  x^  +  2hx  -h  ex.  7.  -'2hm-'Sbrfi  +  4i  hp. 

3.  a^  +  2ah  +  ac.  8.  —ilac  +  99  acd. 

4.  2  ax -2  ay  +  2  a\  9.  4  a^cZ  -  4  ac^  +  12  c^d'. 

5.  3  a3  -  3  a5(?  +  3  ad.  10.  2  a3a;2  +  2  aV  -  6  a;3^/2^ 

11.  Z^y  -Z  2:^  +  9  ax\ 

12.  5  mo;  —  15  mnx  —  5  7ia;  +  X^px. 

13.  14  a3a;y  +  7  a26xY  +  49  alH^y^  -  21  S^^^^ 

14.  91  aid  +  21  52^  -  7  cJ  +  14  ^^. 

15.  2  ^46^2-  2  a4J3^2^  10^263^2, 

16.  -  ^262,^4  +  ^35^.4  +  ^J3^4  „  ^5^6, 

17.  3  a^7?  -  9  ac2:4  +  15  d?x^  +  6  2^^. 

18.  3  a66-a562  + 2^453 -4^255. 

19.  4  w2  +  10  mn  +  26  mr?  +  108  ttz^, 

20.  4c?'(i2__4^6^4_12^5J3^.4^5^4. 

21.  5  ^26-- 10  aJ2_  15  J3_  20  a^. 
22c  2  a22:2  —  3  aar^— 5  (j^x—  6  a22;3, 
33.    6  m^  —  3  rrfin  +  12  mn2  —  27  r^. 

24.  a;4^2  __  4  rji^y  ^  g  ^2^2  __  4  ^^,3  _j.  ^^4^ 

25.  12  a%\^  -  24  a^lc^  -  36  aJ^c'  -  4  a  A 

26.  14  a;3^4  _  91  r^yh  _  56  ^4^2^7  _  98  xH\ 


Ch.  VI,  §  85]  FACTORING  77 

CASE  II 

85.  When  the  expression  is  in  the  type  form  a*  ±2ab  +  b^. 

Since,  §  73,  (a  +  6)2  =  aH  2  a6  +  ^2,  the  factors  of  a2  +  2  ah 
+  J.2  are  determined  by  inspection  to  be  (a  +  5)(<Jf  ■\-  J)- 

Similarly,  §  74,  the  factors  of  a'^—2ab-\- h^  are  (a  —  6) 
(a -6). 

1,  Factor  a;^  +  4  x  +  4. 

a;2  +  4  a;  +  4  =  (a!+ 2)(a;  +  2). 

2.  Factor  169  a%^  -2Qah  + 1. 

169  a262-  26  a/>  + 1  =  (13  a6  - 1)(13  db  - 1). 

EXERCISE  XXXVI 

Factor : 

1.  2? +  14  a;  4- 49,  9.  ^Ix^-IUxy  +  imy^ 

2.  a:2_l4a;  +  49.  10.  484  a:*  -  44  a;2y2  +  ,^, 

3.  a;6-4a;3  +  4.  11.  256  a;2«/2-96<?a:y+9c2. 

4.  9a;2_24a;  +  16.  12.  4'^  a^«/^+ 14  ^2^;^ -|.  d*. 

5.  4  a;2  _  20  a;«,r  +  25  2/2.  13.  144  c2  -  24  <?  +  1. 

6.  25  a2  +  70  a6  +  49  52.  14.  1  +  28  m2  + 196  m*. 

7.  36a*-84a2c  +  49c2.  15.  49  x*  -  28  a^  +  4. 

8.  289  a^+ 136  a;?/ +  16  ^2.  ig.  81  w4  +  144»j3+64  jr?. 

17.  25  m*  + 130  m2n  + 169  w2. 

18.  (a  +  a;)2-2(a  +  a;)  +  l. 

19.  l-4(a  +  J)+4(a  +  6)2. 

20.  (a  +  6)2-2(a  +  S)((?  +  t?)  +  (c  +  c?)2. 

21.  (a; -  yy  -  2{x  -  y)(y  -  z)  +  (y  -  sy. 

22.  9Ca-J)2-6(a-6)  +  l. 


78  ELEMENTARY  ALGEBRA  [Ch.  VI,  §  8( 

CASE  in 

86.  When  the  expression  is  in  the  type  form  a^  —  b\ 

Since,  §  75,  (a  +  b^(a  —  b)  =  aP'—lP;  the  factors  of  a^—h 
are  determined  by  inspection  to  be  (a  +  6)(a  —  6}. 

1.  Factor  a^  —  49. 

a;*-49  =  (a,-2  +  7)(a^-7). 

2.  Factor  (a  -  5)2  -  c^. 

(a -  6)2 _  (c)2  =  |(a - 6)  +  cH(a - 6)  - c|, 
=  (a  —  b  +  c)(a  —  b  —  c). 

EXERCISE  XXXVII 

Factor : 

1.  X^-IU.  12.     (a-6)2-OT2. 

2.  16  a2- 121  62.  13.  (a-by-(c  +  dy. 

3.  1-I00c2.  14.  l-(a-6)2. 

4.  l-196«/2.  15.  (a_6)2_l. 

5.  x*-25z/\  16.  25-(a2_5)2. 

6.  9a2_4  62.  17.  (am +c2)2- 36. 

7.  64a262_i.  18.  a*-(c2-<i2)2, 

8.  25a^-4^2.  19.  (a2-Jc)2-(a-<;)2. 

9.  49w2-16a;y.  20.  (m  -  n2)2  -  (^2  +  ^2)2. 
.10.  121  52a;2^2  _  225  g2.  21.  Ca  +  b  +  cy-(c+d  +  ey 

XX.   d^-ib-cf.  22.    (a  +  J-c-«6c)2-l. 

23.  (a2  +  2  6c-c?2)2_(2a;  +  8^)2. 

24.  (5c2-7acZ)2-121J2. 

25.  64a262-(7(Z  +  lle)2. 

26.  Qfi  +  m2  +  w2^2  _  (-  _  53  _  c3  _  (^3)2. 


Oh.  VI,  §  87j  FACTORING  79 

87.  The  terms  of  an  expression  may  sometimes  be  ar- 
ranged to  show  the  type  form  a^  —  V^, 

1.  Factor     d^ -^ab +  }fi  -  <? -\-1cd- ^. 

=  (a^ - 2  a&  +  y')  -{(?-2cd  +  d^, 

^\(a-h)  +  {c-d)\\(a-  b)-(c  -  d)U 
=  (a  —  b  +  c  —  d)  (a  —  b  —  c  +  d). 

2.  Factor  m*  +  ^^  -  ^2  -  J2  +  2  m^n^  +  2  ab. 
m^  +  n^-a'-b^  +  2mhi^+2ab 

=  m^  +  2  771^71^  +  n'^'—a^  +  2ab  —  b', 

=  (m'  +  2  mhi'  +  n')  -(a'-2ab  +  b% 

^(m?  +  ny-{a-b)\ 

=  l(m'  +  V?)  -f  (a-  b)\  \{m?  +  n") -  (a  -  6)|, 

=  (w?  +  n^  +  a  —  b)  (w?  +  n^  —  a  +  b). 

3.  Factor  ^x^-l  +  'd y^ -IQ  rrfi -12xy +  ^m. 

4a;2_l +  9 2/2-16  m2-12a;2/  +  8m 

=  4aj2_i2a;?/  +  9  2/2-16m2+8m-l, 

=  (4  aj2-12  xy+^  ^')-(16  m^-S  m+1), 

=  (2a?-32/)'-(4m-l)2, 

=  K2a;-32/)  +  (4m-l)n(2^-32/)-(4m-l)|, 

=  (2  a?- 3  2/ +  4  m  - 1)(2  a:- 3  2/ -  4  m  + 1). 

It  should  be  noticed  that  the  terms  containing  cross 
products  show  the  order  in  which  the  terms  should  be 
grouped. 


80  ELEMENTARY  ALGEBRA  [Ch.  VI,  §  8' 

EXERCISE  XXXVIII 

Factor : 

1.  a^-2ah  +  P-l.  s.  9-Trfi-{2mn-n^. 

2.  l-4a  +  4a2-a;2.  6.  a^  +  9-Qx-2da\ 

3.  x'^  +  f+2xt/-9t».  7.  16a^  +  8a;-100a2  +  l, 

4.  l-a^  +  2x7/-i/\  8.  l-6a-4952  +  9a2. 

9.  9a2_9  62  +  42J_49. 

10.  4a^-lQa%^-20xt/  +  25i/^. 

11.  a'^+2ah  +  P-<^  +  2cd-cP. 

12.  c^-2ed  +  d'^-a^  +  2ah-b\ 

13.  ei  +  2cd  +  d^-a^-2ab-b^. 

14.  4:a?-4:xy  +  i/^-a'^  +  2ah-^. 

15.  rr*-62;2y  +  9^2_9a2  +  30ad2-25(^. 

16.  4nfl  +  20  mn  +  25n^-d  c^- 12  cd- 4  d^. 

17.  a^  +  2a?y2  +  ^4_49c2  +  i4c_i. 

18.  l  +  4:X  +  4:X^-X^-4:X^t/—4:y^. 

19.  25-49c*-10a;2y  +  28c2cZ  +  a;V-4(P. 

20.  l-4a;y-c*-<Z2g24.4a;2^2_2c2(;e, 

21.  4f-l-25x^  +  10x-12^z  +  9z\ 

22.  1  -  9  z2  _  a;4  +  16  x^f  -  6  0:%  -  8  a;y. 

23.  -6a?2/2  +  9/-92!*  +  a:*-a;2/  +  6  32xy. 

24.  4  a2  +  9  J22;2  _  16  ^4^  -  9  52  +  24  a25a;2  _  12  ahx. 

25.  49  w2a;2  -  169  m^ai^  -  16  «2j^2  ^  25  »w2/  + 104  mnx^ij 

—  70  mnxy. 


Ch.  VI,  §  88]  FACTORING  81 

88.  An  expression  in  the  form  a*  +  a-b^  +  b*  may  be  said 
to  be  in  the  disguised  form  of  the  difference  of  two  perfect 
squares,  and  may  be  factored  as  before. 

1.  Factor  a*  +  a262  4. 54. 

a*  +  a-b^  +  6*  =  a*  +  a'l^  +  b*  +  (a'b^  -  aV), 
=  a^  +  2a'b-  +  b*-a'b% 
=  (a^  +  b'-y-{aby, 
=  {a'  +  ab  +  b')(a'-ab  +  b^. 

2.  Factors* +  9 0^  +  81. 

a;*  +  9a^  +  81  =  a;*  +  9a^  +  81  +  (9a^-9a^), 
=  (a!^  +  18a;2  +  81)-(9a^), 
=  (a;2  +  9)2_(3a;)2, 

=  (x'  +  3x  +  9)(x'-3x  +  9). 

3.  Factor   a*- 13^2 +  4. 

a<_13a2  +  4  =  a*-4a2  +  4-9a^ 
=  (a^-2f-(3ay, 
=  \(a'-2)+3a\l(a'-2)-3al, 
=  (a^  +  3a-2Xa^-3a-2). 

EXEBCISE  XXXIX 

Factor : 

1.  a* +  a^b^  + 25  bK  6.  a;*  + 11  a^y  +  36  «/*, 

2.  l  +  m^  +  m*.  7,  a*  +  a^  +  l. 

3.  4a;*-6l2;y  +  9/.  8.  9a*-16aW  +  4bK 

4.  a;*  +  a;y  +  2/*-  9-  a;*  -  8  a^«/2  +  4 /. 

5.  a*  +  9a.2  +  81.  10.  625  a*  +  25  a^  +  1. 


82  ELEMENTARY  ALGEBRA  [Ch.  VI,  §  89 

CASE   lY 

89.  When  the  expression  is  in  the  type  form 
a'  +  b'-\-c^  +  2ab  +  2ac  +  2bc. 

Since,  §  77,  (a  +  J  +  c)2  =  a2  +  62  +  c2  +  2  a6  +  2  ac  +  2  5c, 
the  factors  of  a^  +  b^+<^+2ab  +  2ac+2bc  are  determined 
by  inspection  to  be  (a  +  6  +  c)(a  +  J  +  c). 

1.    Factor  a'^  +  P  +  e^-2ab +  2  ac- 2  be. 

The  cross  products,  —2ab  and  —  2  be,  in  both  of  which  6 
occurs,  show  that  b  has  the  minus  sign. 

a^  +  b^'  +  c^-2ab  +  2ac-2bc  =  (a-b  +  c){a-b  +  c). 

EXERCISE  XL 
Factor : 

1.  a^  +  b^  +  c^  +  2ab-2ac-2bc. 

2.  a^  +  b^  +  c^-2ab-2ac  +  2  be. 

3.  (fi  +  P  +  (^-2ab  +  2ae-2be. 

4.  x^  +  4:i/^  +  9  z^  —  4:xi/  —  Gxz  +  12  yz. 

5.  4  a2  4-  9  J2  +  ^2  _  12  a6  -  4  ac  +  6  be. 

6.  25  a^  +  4  a;y  +  y*  -  20  a^y  + 10  x^y^-  4  xy^. 

7.  4  a;2  +  9  a;2a2  +  16  ^252  -i2x^a  +  16  x%  -  24  x^ah. 

8.  16  52;c2  +  a*  +  4  J2  + 16  J22;  _  8  d^bx  -  4  a25. 

9.  4  a;*  +  a::2?/2  ^  9  ^2  _  4  3^3^  ^  12  a;2y  _  g  xy^. 

10.  25  e*  +  30  ahc"^  -  20  ac^  +  9  a2j2  _  12  a%c  +  4  ^2^2. 

11.  12  a62c8  +  4  6*52  +  ^452  _  4  ^2^3^  _  6  a%<P'  +  9  d^a^. 

12.  60  a3J3  +  1  _  10  a2f  _  12  aJ2  +  36  aSJ*  +  25  a*b\ 


Cii.  VI,  §  90J  FACTORING  83 

CASE  V 

90.  When  the  expression  is  in  the  type  form 

Since,  §  78,  (a  ±by  =  a3±8  o?h  +  3  aS^  ±  J^,  the  factors 
of.  a^  ±  3  a^J  +  3  ah"^  ±  W  are  determined  by  inspection  to 
be  (a±  J)(«±6)(a±5). 

1.  Factor  a?  —  Z  x^y  +  3  xy"^  —  y^. 

^-^^y-^'ixf-f  =  {x-y){x-y){x-y)Jy^'nJ 

2.  Factor  l-6a;+12a;2-8aj3. 

1  _  6  a;  + 12  ic^  -  8  a^  =  1  -  3  (1)2(2  a;)  +  3(1)  (2  a;)2  -  (2  a;)', 
=  (1  -  2  a;)(l  -  2  a;)(l  -  2  a;). 

EXBECISB  XLI 

F-actor : 

1.  TO^  — 3  m^w  +  3  mn^  — w^.  3.   a3  +  3a2  +  3a  +  l. 

2.  \-Zy-\-%y'^-y^.  4.    %  a^-Vla^  +  <o  a-\, 

5.  27a^-54a;*5  +  36a;262_8  63.   . 

6.  125a3  +  75a2  +  l6a  +  l. 

7.  8  63  +  36  6%  +  54  6c2  +  27  c3, 

8.  l-21a;  +  147a^-843a;3. 

9.  216a3  +  108a22;  +  18aa^  +  a^. 

10.  8a3-36a25  +  54aJ2_27R 

11.  27  a3a:3  +  iQg  a26a;2«/  +  144  ahHy^  +  64  Sy. 

12.  125  x^y^  -  300  aJa;2«/2  +  240  aWxy  -  64  a%^. 

13.  1331a6_l089a*6  +  297a252_27  63. 


84  ELEMENTARY   ALGEBRA  [Ch.  VI,  §  91 

CASE   Yl 

91.  When  the  expression  is  in  the  type  form 

jr^  +  (a  +  b)x  +  ab. 

Since,  §  79,  (x  +  a}(^x  +  b)  =  x^  +  (^a  +  b}x  +  aS,  the  fac- 
tors of  x^  +  (^a  +  b^x  +  ctb  are  determined  by  inspection  to 
be  (x  +  a'){x  +  J). 

1.  Factor  x'^  +  8x  +  15. 

x^  +  Sx  +  15  =  (x  +  3)(x  +  5). 

2.  Factor  x^'-2x-15. 

x^^2x-15  =  (x  +  3)(x-5). 

It  is  to  be  noticed  that  the  algebraic  sum  of  the  factors  of 
the  last  term  is  the  coefficient  of  the  middle  term  in  the  tri- 
nomial, and  it  is  therefore  necessary  to  find  two  numbers  such 
that  their  product  is  the  last  term  and  their  sum  the  coefficient 
of  the  middle  term. 

EXERCISE   XLII 

Factor : 

1.  x^  +  5x  +  6.  5.  0:2-3^-10.  9.  :i:2__8  2;  +  7. 

2.  x^-2x-S.  6.  x'^-3x-28.  10.  x^-5x-S6. 

3.  x^-4:X-5.  7.  x'^  +  5x-24.  ll.  a'^  +  2a-S5^ 

4.  x^—bx  +  4:.  8.  x^  —  x—20.  12.  a^  —  1  a  —  l'i. 

13.  a^  -  8  a2  -  33.  19.  x^  +  bx  +  cx^  be, 

14.  a^ +  12^2  + 11.  20.  x^  —  bx  +  cx  —  bc, 

15.  a^  —  11a  —  42.  21.  x^  —  bx—  cx  +  be, 

16.  a^J^  _  9  ^2j2  __  136.  22.  i?:2  +  5:r  -  ^2;  -  5^. 

17.  ^2  -  26  a  +  133.  23.  a;2  +  (a  -  c?)^;  -  6X^. 
18.-  aV--19aV-92.  24.  ^  +  (a  +  ^)2;  +  ai. 


Ch.  VI,  §  92]  FACTORING  86 

CASE   VII 

92.  When  the  expression  is  in  the  type  form 

Since,  §  80,  (ax  +  h^(cx  +  d)==  acx^  +  x(lc  +  ad^  +  hd^ 
the  factors  of  acx^  +  x(hc  +  ad)  +  Id  are  determined  by- 
inspection  to  be  (ax  +  V){cx  +  c?). 

1.    Factor  1x^ +  bxy -Zy^, 

2  x^  -\-h  xy  -Z  y"  =  {2  x-y){x  +  ?>  y). 

Since  the  first  term  of  the  trinomial  is  the  product  of  the 
first  terms  of  the  binomials,  the  first  terms  of  the  binomials 
must  be  2  a;  and  x ;  since  the  last  term  of  the  trinomial  is  the 
product  of  the  last  terms  of  the  binomials,  the  last  terms  of  the 
binomials  must  be  3  2/  and  y.  The  sign  of  the  last  term  of 
the  trinomial  is  minus ;  hence  the  last  terms  of  the  binomials 
must  have  opposite  signs.  By  trial  the  factors  are  now  found 
as  given  above. 

If  the  trinomial  contains  no  common  monomial  factor, 
the  binomial  contains  no  common  monomial  factor. 

The  middle  term  is  found  by  multiplying  the  first  term  of  the 
first  binomial  by  the  second  term  of  the  second  binomial,  ar  ^ 
by  multiplying  the  second  term  of  the  first  binomial  by  the  ^ 
term  of  the  second  binomial,  and  taking  the  algebraic  sum  . 
these  products  for  the  middle  term.    The  process  is  represented: 


2x^  +  ^xy-?>f={2x-y)(x^2>y). 

Writing  the  possible  factors  of  2  oi?^ 

2x^  +  5xy-3y'^        (2  x        )(x        ), 

and  in  the  parentheses  writing  also  the  possible  factors  of 
—  32/^,  the  factors  of  2xi^  +  5xy  —  3y^  are 


86  ELEMENTARY   ALGEBRA  [Ch.  VI,  §  92 

either  (2^  +  3?/)  (x  -  y),  (1) 

or  (2x-3y){x  +  y),  (2) 

or  (2x  +  y)(x-3y\  (3) 

or  (2x-y)(x  +  3y).  (4) 

Each,  of  the  possibilities  (1),  (2),  (3),  (4)  must  be  tried  by 
actual  multiplication  until  the  proper  factors  are  discovered. 

2.    Factor  2^:2 +  5  2:2/ -12/. 

The  possible  factors  of  2  x^  are  2  x  and  x ;  the  possible  factors 
of  12  y^  are  12  y  and  y,  y  and  12  ?/,  6y  and  2  y,  2  y  and  6y,  4:y 
and  Sy,3y  and  4  ?/.  That  is,  2  a;  and  a?  must  be  tried  with  each 
of  the  six  possible  factors  of  12  y^. 


(2x 

12  2/)  (X 

y)> 

(1) 

{2x 

y)(x 

12  2/), 

(2) 

{2x 

6y)(x 

2  2/), 

(3) 

(2x 

2y){x 

6  2/), 

(4) 

(2x 

4.y)(x 

Sy), 

(5) 

(2x 

3y)(x 

iy)- 

(6) 

Possibilities  (1),  (3),  (4),  and  (5)  are  immediately  eliminated 
because  the  binomials  contain  a  factor  which  is  not  a  factor  of 
the  trinomial.     By  trial, 

2 x'  +  5  xy  ^12 y'  =  (2  X-  3  y)(x  +  4:y). 


/ 


3.    Factor  2  a;2  -  <a  +  2  5)  +  ^5. 


The  possible  factors  are : 

(2x 

a)(x 

H 

(1) 

(2x 

h)(x 

a), 

(2) 

(2x 

ab)(x 

1), 

(3) 

{2x 

l)(x 

ah), 

(4) 

2x^ -x{a  +  2h)  -\- ah  =  {2 X-  a){x -h). 


Ch.  VJ 

,  §  92]                             FACTORING                                              8' 

ir-li    -^ 

EXERCISE  XLIII 

Factor  : 

i        1. 

6a;2  +  7:r  +  2. 

18. 

18:r2-x-39. 

L    2. 

4:X^+8x  +  3. 

19. 

10x2-59x  +  85. 

3. 

2x^-Sx  +  l. 

20. 

95x2-138x  +  7. 

\     4. 

Sx^-8x  +  4:. 

21. 

16x2-211x+39. 

5. 

Ga^  —  x  —  l. 

22. 

24x2-54x-105. 

6. 

2x^  +  3x-2. 

23. 

84x2  +  148x-112. 

7. 

Sx^-8x  +  5. 

24. 

15x2-101x^-28. 

8. 

^x^  +  x-^lX. 

25. 

6  x2  —  61 X  —  55. 

9. 

3:c2  +  lla;-  20. 

26. 

60x2-147x-156- 

10. 

8  2^2 +  10  2; -12. 

27. 

6a;2-77x+92. 

11. 

62;2-72;-20. 

28. 

80x2-28x-52. 

12. 

32;2-29  2:  +  40. 

29. 

10x2  +  9x-91. 

13. 

16:^2_82  2;-33. 

30. 

84x2-158x  +  70. 

14. 

26x2 -141 2: -11. 

31. 

aJx2  —  X  (6  +  ac)  +  c. 

15. 

34x2  +  131^-99. 

32. 

ax2  +  x(l  +  ^J)  +  &. 

16. 

30x2-9a;~3. 

33. 

a5x2  +  X  {ad  —  he)  —  cd. 

17. 

28x2  +  23^-15. 

34. 

1aH^^-x{2ac-aV)-lc. 

35.  «Jx2  +  x{ac  —  52)  _  Ic. 

36.  aJx2  —  x((22  _|_  5^^  _^  ^^^ 

37.  2  aJx2  -  xQ)G  +  4  (^2)  +  2  a^. 

38.  mn:x?  —  x(an  —  3  hm)  —  3  a6. 

39.  2ax2-x(3a(^  +  4^)+6ccZ. 

40.  7ax2-x(a-14cZ)-2df. 

41.  8  mnx2  —  X  (12  a/i  —  10  ItyC)  —  15  ah. 


88  ELEMENTAllY  ALGEBRA  [Ch.  VI,  5  S3 

CASE   VIII 

93.   When  the  expression  is  in  the  type  form  a'  ±  b^. 

By  §81,       a^-^¥=^{a^-ab  +  }P-){a  +  h-). 
a^-lfi=-.  (a2  +  a6  +  W-^ia  -  6). 

Hence  the  factors  of  0?  +  W"  are  determined  by  inspection 
to  be  (a  +  V)  {c?  —  ah  +  6^) ;  and  the  factors  of  0?  —  W"  are 

1.  Factor  a^  +  'f: 

2.  Factor  a^  + 27. 

=  {x  +  Z)i(xf-{x){S)  +  (Sf], 
=  (a;  +  3)(a^-3a;  +  9). 

3.  Factor  125  2^3  _  64  g3, 
125  ar' -  64  z»  =  (5  a;)3  -  (4  z)', 

=  (6  a;  -  4  2)  [(5  a;)^  +  (5  a;)  (4  z)  +  (4  z)«], 
=  (6  a;  -  4  z)  (25  a^  +  20  as;  +  16  2^). 

4.  Factor  a?  +  y^. 

=  [_x  +  f\[_{xy-{x){y'^  +  {yyi 
—  (x  +  /)  (a;2  —  xy^'  +  y*). 

5.  Factor  64  -  (a  -  hy. 

64  -  (a  -  6)5  =  (4)3  -  (a  -  &)", 

=  [4  -  (a  -  6)]  [(4)^  +  4(a  -&)+(«-  6)^], 
=  (4  _  a  +  6)  (16  +  4  a  -  4  &  +  a^' -  2  a6  +  6*; 


Ch.  VI 

,  §  y3J 

FACTORING 

Factor: 

EXERCISE  XLIV 

1. 

oi?-y^. 

12. 

2/3-27^. 

2. 

x^  +  yK 

13. 

1  - 125  a3. 

3. 

%  +  aK 

14. 

216a;y +  27J»• 

4. 

a3_8. 

15. 

a3J6-,.l2. 

5. 

3^  +  64. 

16. 

64  ^3  _  53. 

6. 

126  a3  _ 

8R 

17. 

1  -  343  h\ 

7. 

8  63- 125  a3. 

18. 

729m3-l. 

8. 

343  6-3  - 

1. 

19. 

1000^*3  +  1. 

9. 

1  +  1000&3. 

20. 

7?^^  —  (^  —  J)^ 

10. 

512(^3- 

a;i6. 

21. 

(a  +  hy  +  (?. 

11. 

8^  +  yS 

'. 

22. 

(jn^ny+ia  +  h)\ 

89 


23.  8(a2_J2)3_l. 

24.  27(a-c)3  +  64(6-c?)». 

25.  343(c2  +  (^)3_l. 

26.  (6c3-«3)3_(a2  +  52-)8, 

27.  (s  +  a)3— (s— c)3. 

28.  1728(a;+«/)3-343(a;-2^)«. 

29.  729  (a  -  6)3  4- 125  («  + 6)3. 

30.  512(a  +  6  +  c)3-1331(a-6-e)«. 

31.  a;3(a  _  6  +  c)3  +  a3(-2.  —  y  +  2)8. 

32.  ofi{a  +  6  +  c)3  -  63(a;  +  2/  +  z)^. 

33.  «6J3^^2  +  ^  _  2)3  +  ^359(^3.  _  ^  _  22-)8. 

34.  0(3(^2;  +  y  —  2)3  —  63(^y  +  2  —  a;)*. 

35.  343  aXs  +  a  -  6)8  + 1000  b^Qa  +  b  +  s)». 


90  ELEMENTARY  ALGEBRA  [Ch.  VI,  §  94 

CASE  IX 

94.    When  the  terms  of  the  expression  can  be  arranged 
to  show  a  common  binomial  factor. 

1.    Factor  ax  +  ay  +  bx  +  hy. 

ax  +  ay  +  bx  +  by  =  (ax  +  ay)  +  (hx  +  by)^ 

=  a(x  +  y)  +  b{x  +  y), 

=  (a  +  b)(x+y). 

After  the  terms  have  been  arranged  to  show  the  common 
binomial  factor,  the  process  is  really  one  of  division,  thus : 


^(^  +  y)  +  K^  +  y) 
ct(^-^y) 


(^  +  y) 


a  +  b 


b(x  +  y) 
b{x  +  y) 

Case  IX  may  then  be  considered  as  an  extension  of  Case  I; 
in  Case  I,  a  common  monomial  factor  is  removed ;  in  Case  IXe 
a  common  binomial  factor  is  removed. 

2.  YdiQiov2a?-?>ax  +  4tlx^-Qal. 

2  a;3  _  3  ^^  +  4  5^  _  6  a&  =  (2  ar^  -  3  aoj)  +  (4  &a;2  -  6  ab\ 

=  a;(2  a;2  -  3  a)  +  2  &(2  i»2  -  3  a), 
=  (x  +  2  2>)(2aj2-3a). 

3.  Factor  2  0^-4:1:2^ -3  a: +  6  2/. 
2^-4.x'y-^x  +  Q>y=(2o?-A.x'y)-{^x-Qy), 

=  2a^(x-2y)-S(x-^2y), 
=  (2x''-3)(x-'2y). 

4.  Factor  12  a%^  -  42  h^c  +  16  a^  -  56  ahc. 

12  a^b^  -  42  b'c  + 16  a^  -  56  abc 

=  2  [3  b\2  a^  -  7  &c)  +  4  a(2  a^ -  7  5c)], 
=  2(3  62  +  4a)(2a2~7  5c). 


Ch.  VI,  §  94]  FACTORING  91 

EXERCISE  XLV 

Factor : 

1.  mx  +  am  +  nx  +  an, 

2.  mx  —  am  ■—nx  +  an. 

3.  2  hx^  —  ahx  +  4:Cx—2  ac. 

4.  m^n  —  3  abn  —  2  m?^  +  6  ahp. 

5.  x^  +  ax  +  bx  +  ah. 

6.  x^  —  ax  —  hx  +  ah, 

7.  6  hx  —  15  ah  —  4:  dx  +  10  ad, 

8.  dmnx  —  ac^dx  +  mnrs  —  ac^rs, 

9.  —2an  +  Sap  +  2hn  —  S  hp. 

10.  6  eg  —  9  c?^  +  4  a^?  —  6  acZ. 

11.  a4i2  -  2  a264  +  ^^5  -  2  aS^. 

12.  2  5(?m  —  4  ah^c  +  7  am^i  — 14  a%n. 

13.  2  T/i^/i  —  2a'^h  —  8  cm^n  +  3  a^J^?. 

14.  —  a%x^  —  4  Je  —  2  ^2^2^  —  S  cy. 

15.  8  aa2  +  10  art  +  12  6s2  +  15  hrt. 

16.  —  mnx  —  2  mn  +  p^x  +  2  jt?^. 

17.  14  a^e^f  +  35  ^2^62^  _,.  q  ^2^^^  +  15  52^^^^ 

18.  10  a(?  +  Jc  -  110  ad  -  11  5c?. 

r 

19.  rs  +  a^w  —  3  c^rs  —  8  a^t^n. 

20.  8  acajy  —  14  o^xz  +  21  acc^s  — 12  c^cZy. 

21.  6  a3  -  8  a262  _  15  ^5^  +  20  Jfic. 

22.  6  a^  -  33  aca;  -  8  ca:^  +  44  ^^2. 


92  ELEMENTARl    ALGEBRA       [Ch.  VI,  §§  95,  96 

95.  A  theorem  is  a  statement  of  a  general  truth  which 
requires  demonstration. 

THE  FACTOR   THEOREM  I 

96.  If  any  expression  containing  x  reduces  to  0  when  a  is 
substituted  for  x,  then  x—  a  is  a  factor  of  that  expression. 

Let  E  represent  the  expression.  Divide  E  by  x  —  a 
until  the  remainder  does  not  contain  any  power  of  x.  Let 
R  be  the  remainder,  and  Q  be  the  quotient.     Then 

E=  QQc-^a^  +  R.  (1) 

Equation  (1)  is  always  true  whatever  may  be  the  value 
of  x.     Take  x  =  a^  and  substitute  in  (1)  ; 

0=^(a-a)  +  i2,  (2)| 

0=^(0)  +  i2,  (3) 

0  =  72.  (4) 

In  (2),  E  becomes  0,  because  the  expression  is  taken  as 
one  containing  x^  which  becomes  0  when  a  is  substituted 
fora; ;  also  in  (2),  a  —  a  =  0,  and  ^0  =  0.  Li  (4),  R  be- 
comes 0  ;  or,  in  other  words,  there  is  no  remainder.  Con- 
sequently a;  —  «  is  an  exact  divisor  or  factor  of  E. 

Since  a;  + a  =  2;— (— a),  the  theorem  holds  true  if  (a) 
be  replaced  by  (— ^)  in  the  statement,  thus:  if  any  ex- 
pression containing  x  reduces  to  0  when  (— <^)  is  substi- 
tuted for  Xy  then  a;—  (—a),  or  x  +  a^is  a  factor  of  that 
expression. 

The  Factor  Theorem  has  a  wide  application,  and  maj 
be  applied  as  a  check  to  most  of  the  preceding  cases,  and 
to  many  forms  which  are  not  included  in  those  alreadj 
iiyivea 


Cii.  VI,  §  9b']  FACTORING  93 

1.  Factor  a^  +  1. 

Take  x  =  l,  and  substitute  in  o;^  + 1 :  1^  + 1  =  1  + 1  =  2. 
.'  I  ere  the  expression  oc^  +  1  does  not  become  0,  or  vanish,  and 
.s( )  cc  —  1  is  not  a  factor.  Take  x=  —1,  and  substitute  in  ar^  + 1 : 
(  - 1)^  + 1  =  —  1  + 1  =  0.  The  expression  vanishes,  and  x  +  1 
is,  therefore,  a  factor  of  a?^  +  l.  By  division  the  other  factoi 
or  factors  may  be  established. 

2.  Factor  afi  +  y^. 

Take  x  =  ?/,  and  substitute :  (yY  +  y^  =  2y^;  ^  —  y  is  not  a 
factor.     Substitute  x—  —y:  (—  ?/)^  +  2/^  =  0 ;  a;  +  2/  is  a  factor. 

^  j^ 'if  z=  {x  +  y)  (xl^  —  Q(?y  +x^y^  —  xy^  +  y^)o 

The   second   factor,   x'^^ii(?y +  Q(?y'^ —  xy^  +  y^^  is    found  by 

division. 

3.  Factor  :i:5  +  32 /. 

By  substitution,  x  +  y  and  x  —  y  are  shown  not  to  be  factors. 
Try  x  =  2>  y  '.  (2 yy  +  ^2  if  =  Q>4.y^',  x  —  2y  is  not  a  factor.  Try 
i7;  =:  —  2  7/ :  (—  2  2/)^  +  32  ?/^  =  0 ;  cc  +  2 1/  is  a  factor.  By  division 
x^  —  2  x^y  +  4  xPy^  —  8  a?^/^  +  16y^  is  the  other  factor. 

a;^  +  32  2/^=(a:  +  2  2/)(a;*-2a^i/  +  4a^/-8aji/^  +  16^^). 

4o    Factor  :2;3  +  ^_7^  +  2. 

The  substitution  of  x  =  l,  x=  —1  both  give  remainderfe t 
now  try  a;=:2:  (2)^  +  (2)'- 7(2)  +  2  =  8 +  4  -  14  +  2=  0  ; 
a;  —  2  is  a  factor. 

By  division,  x^  +  3  a;  —  1  is  the  other  factor. 

ar*  +  ^- 7 aj  + 2  =  (a;- 2)(a;2 _j_ 3  ^ _ j^)^ 

The  factor  obtained  by  division  must  be  carefully 
inspected  to  determine  if  it  is  prime. 


y4  ELEMENTARY  ALGEBRA  [Ch.  VI,  §  96 

EXERCISE  XIiVI 

By  use  of  the  Factor  Theorem,  separate  in  factors: 

1.  ^  +  f.  9.  a2-16a  +  64.  17.  a}-W. 

2.  afi+y^.  10.  a^— 25.  18.  1  — Sa:^^ 

3.  a*-l.  11.  a;2-57a;  +  56.  19.  2l3^-f. 

4.  a^  +  8.  12.  343-a;3.  20.  32a;5+j^. 

5.  1  +  a^.  13.  a--*— 81.  21.  a^  —  y^. 

6.  a^-27.  14.  a;5  +  243.  22.  a^^  +  ^e, 

7.  64-a;3.  15.  82a^+b^  23.  a^^  +  ^/S, 

8.  32 -a^.  16.  a'  +  b''.  24.  x^  +  yo^ 

25.  a^-a;2-a;  +  l.  32.  a;3_^2_  2  a;_  12. 

26.  a:3-a;2-3a;-l.  33.  a^+ 2  a;2- 3  a;+ 20. 

27.  a^-2a^-6a;-2.  34.  3^  +  :^-inx-21. 

28.  a^-3a;-2.  35.  a^-S  x^+12x+ 9. 

29.  a:3-a;2-a;-2.  36.  a^+lB3^+ iBx+ 6. 

30.  a^  +  a^  -  a;  +  2.  37.  ai^+Za^-x^-2x+l. 

31.  a;3_5a;  +  2.  38.  a*-a^+5  a^+14:a-lQ. 

39.  2a^-5a^  +  lSx^-9x-l. 

40.  3a:*+8a^  +  8a^  +  Ta;-2. 

41.  2a4-7a3+8a2_6a  +  4. 

42.  a*  +  Qa^  +  lla^-a-21. 

43.  a*-Sa^  +  n  a^-Ua  +  8. 

44.  2a^-13a^+16a;2-6a;+5. 

45.  3^-4:x^  +  10a^-5x^-4:X+2. 

46.  3a^  +  3a;*  — 5ar^-6a;-4, 


Ch.  VI,  §  97J  FACTORING  95 

HINTS   ON   FACTORING 

97.  It  is  impossible  to  give  any  definite  method  of 
attack  in  factoring.  A  monomial  factor  should  at  once 
be  removed.  Every  factor  should  be  carefully  inspected 
for  further  factors.  It  may  happen  that  an  expression 
can  be  factored  by  different  methods.  If  the  expres- 
sion can  be  factored  as  the  difference  of  two  squares,  it  is 
generally  preferable  to  do  so. 

1.  Factor  a;^  —  2/^. 
By  Case  III, 

by  Case  VIII,  ={x  +  y)  (x^  ~xy  +  y^)  {x  -  y)  {a?  -{-xy  +  y'^). 
By  Factor  Theorem, 

x^  —  y^=  (x  —  y)(x^  +  x'^y  +x^y^  +  x'^y^  +  xy^  +  y^), 

=  (x-y)  lx\x  +  y)  +  x^y\x  +y)  +y\x-]r  y)  ], 
by  Case  IX,      ={x  —  y){x  +  y)  (x^^  +  o?y^  +  y^), 
by  §  88,  =^{x-y){x  +  y)(x^  +  xy  +  y'^(x'--xy  +  fy 

2.  Factor  afi  +  y^. 

By  Case  VIII,  x'  +  y'=  (xy  +  (y'f  =  (x^  +  r)  (x'  -  xhf  +  y'). 

3.  Factor  2:12 +  64. 

By  Case  VIII,  x"'  +  64  =  {xy  +  (4)^  =  (x'  +  4) {x^ -^x'  +16). 

4.  Factor  x^  —  y^. 

By  Case  VIII,  x'-y'=:  (xy  -  (f)^  =  (o^-  f)  (x'  +  x^f+  y% 
=  {x-y)(x'  +  xy  +  f)(x'  +  :^f  +  yy 

5.  VsLCtoT  x'^^  +  y^^. 
By  Factor  Theorem, 


96  ELEMENTARY   ALGEBRA  [Ch.  VI,  §  97 

6.  Factor  3  (a  -  6)3  -  a  +  5. 

S(a-by  -  a  +  b^3(a-  by  -  (a-b), 
by  Case  IX,  =(a-&)[3(a-6)^-l], 

=  (a - 6)  (3  a=-6  a6  +  3  6^- 1). 

7.  Factor  a^  +  b^  +  c^  -  8  abc. 

a3  +  53  ^  c3  _  3  abc  =  (a^  +  6»)  +  (c'  -  3  abc).  (1) 
Now 

a'  +  b^^ia  +  bf-Sa'b-Sab'-^ia  +  by-Sabia  +  b).  (2) 

Substitute  a'  +  6^  =  (a  +  bf  -3ab(a  +  b)  in  (1), 

a3  +  63  +  c'-3«6c  =  (a  +  6)'-3a&  (a  +  6)  4-<^-3  a6c,  (3) 

=  [(a  +  &)*  +c']  -  3  a6  (a  +  &)  -  3  abc,  (4) 
by  Case  VIII,        =(a  +  b  +  c)l(a  +  bf  -c(a  +  b)+  c^J 

-3a6[a  +  6  4-c],  (5) 

by  Case  IX,  =(a+b+c)l(a+by-c(a+b)+c''-3ab2,  (6) 

=  (a  +  6  +  c)  (a^  +  2  ab  +  b^-ac-bc  +  c^ 

-Sab),  (7) 

=  (a  4.  6  +  c)  (a^  +  6^  +  c=  -  ab  -ac-  be).  (8) 

REVIEW  EXERCISE  XL VII 

Factor : 

1.  a;2-22«  +  121.  4.    343:r3-l. 

2.  4a:2_49<j4.  5.   a.4_4a^_60. 

3.  a^  +  28 a;y  +  196  2/2.  6.    9a%^-f. 

7.  l  +  a2  +  62_2a-26  +  2a5. 

8.  l-3a  +  3a2_a3. 

9.    Bx^  +  10xt/  +  Sf.  11.    81  a* -16  5*. 

10.    w*  +  m^n^  +  »*.  12.    a2  _  52  4.  2  Jc  —  c2. 


Ch.  VI,  §  97]  FACTORING  97 

13.  (m  —  n)2  +  2  x(m  —  71)+  0(^. 

14.  a'^x^  +  a^x^  +  l. 

15.    l-T?-y'^-2xy.  18.    (2a  +  hy-(2h  +  ay. 

16     x^-Qax-^h'^-l^ah.      19.    y^-y-Q. 
17.    x^  +  x'^y  +  xy'^  +y^.  20.    6a;2+13a;— 5. 

21.  (27  ^3)2  ~  2  (27  2/^)  (8  J3)  +  (8  63)2. 

22.  1  —  a?x^  —  W'y'^  +  2  ahxy, 

23.  a2  +  J2  +  2aJ-4a262.         38.    2a;3  + 5  2^2- 12a:. 

24.  l-18a;-63:i:2.  39.    27^3  +  125  63. 

25.  ^x^—bx-\-\.  40.    :r2^  +  3  xy^  —  3  a;^  —  ^/^^ 

26.  4  ^2^2  -  (a2  +  62  _  ^2^)2,  41^     56  +  2^  -  ^. 

27.  nx  —x  +  y  —  ny.  42.  5  :?;32/2  _j.  5  ^2^2  _  gO  0:22. 

28.  x^—lQy^.  43.  2;3  +  :i;2  +  ^_|.l^ 

29.  ^^62  -  6  ^35  +  9  ^2,  44.  a:2- 18  0^  +  32. 

30.  ^'ix^y  +  Q^pfiy-QOx^y.  45.  a;2- 2^2;- 2  6:1; +  4 ^6. 

31.  0:4+/- 18^:2^2.  46.  a;^  +  2 :z;2/^2  +  ^2^4. 

32.  3:2;3+2a;2-2:r-l.  47.  aH^-1acxz-h^y'^  +  cH\ 

33.  (7a:  +  42/)2-(2a:-?/)2.  48.  a^  —  6^ 

34.  250(a-6)3+2.  49.  a3  +  63  +  a  +  6. 

35.  ^252  _  ^2  _  52  4. 1,  50.  a3- 2^26 +  2^62-63. 

36.  x^yH'^-x^z-y'^zV^.  51.    (5  a;  -  2)2  -  (a;  -  4)2. 

37.  3a;2--6a:  +  9.  52.    l-^lx- {a^  ^^  aV):^. 

53.  a2a^ — a2/ — 62a;2 + 622/2. 

54.  4(a6  +  (?cZ)2-(a2  +  62-c?2-(;2)2. 

^5.   a3-5a26  +  453,  5g.   64(^^-X. 


96  ELEMENTARY  ALGEBRA  [Ch.  VI,  §  97 

57.  y^  —  c^+2  ex  —  x^. 

58.  be(b  —  t?)  +  ea(^c  —  a)  +  ab(^a  —  6). 

59.  x^—2x'^y  +  x^  —  4:x+8y  —  4:. 

60.  a^-a^b  +  ab^-bK 

61.  a\b  -c}  +  b\c  -  a)  +  c^^a  -  5). 

62.  a;i2-yi2. 

63.  a:^  +  y^^.  70.  a*  —  a^«/  +  az^  —  yz^. 

64.  \\S)--x-x\  71.  a;4-(a;-6)2. 

65.  a{?a:2  —  bcx  +  adJa:  —  bd,  72.  m^  —  64  v^. 

66.  a^J  +  8  a(?bm^.  73.  a^  —  x^. 

67.  4:0^x^  +  4  (P'xy  +  (?^/2.  74.  8  (a:  +  ?/)3  -  (2  a;  -  ?/)''^. 

68.  9  2;^-40a;y +  16y.  75.  ^ -1  x^  ■^Wx-'i. 

69.  a^  +  y^.  76.  3  2j2  +  ^(3^_^5)+^j^ 

77.  a;^  — 5  2;22/2  +  4  2/^. 

78.  (a;3  +  ^_l)2_(;^^_^_l)2, 

79.  x'^  +  Sa^-x^-Sx.  81.    x^-Vy^. 

80.  ai<^  +  6i^  82.    a;3_3^_2. 

83.  (a-2  6)2-9-3(a-2&  +  3). 

84.  d^  —  b'^  +  x^  —  y'^  +  2  (ax  —  by}. 

85.    mx^  +  anx^  —  Pnx  —  ab^m  +  na^  +  amx^  —  b^mx  —  ab^n. 

86.  a;4  +  2a;3  +  2:r2  +  2a;  +  l. 

87.  2:r^  +  a;3  +  4:i:2+6a;  +  2. 

88.  m^  —  7?i^^  —  m^z  —  mz  —  z  —  1. 

89.  2  2:2  +  3  a:,y  +  ^2  __  2  rr  —  2/- 

90.  9  (a2  -  a(?)2- 6  ac?2(a -(?)  +  (?*. 


Cii.  VI,  §  97]  FACTORING  99 

91.  {a-h)(2x^-'2xy)  +  (h-a^(^^xy-2y^). 

92.  (x  -V){x^  2Xx  -  3)  +  (:^  -  1)(^  -  2)  -  (:i;  -  1). 

93.  2;3-2:i:-21. 

94.  x'^—2xy—dy^  —  x^  +  2  x^y  +  3  xy^. 

95.  <ix  +  yy  +  x  +  y, 

96.  (m  +  M  -  3)2  -  3  (m  +  ^  -  3)  —  4. 

97.  x^  —  y^  —  (x  —  yy, 

98.  (a2-62)+^^(3J  +  ^)_2a;Y, 

99.  m^  —  3  m^n  +  3  mn2  —  ?^^  —  m  +  r^. 

100.  x^  +  y^  +  z^—S  xyz. 

101.  (x  —  ^)3  +  1  —  3  (a;  —  2/  +  1). 

102.  x^+  x^+  x^y^  +  x^+  y^+  xy\ 

103.  (x  +  2/)*  —  a:*  —  y\ 

104.  (a3-S3)_(a2_62>^-(a_J)2. 

105.  ^4  +  J4  ^  ^4  _  2  ^2J2  _  2  a2c?2  +  2  62^2. 

106.  4J4c4-J4_2  52^2_^4, 

107.  x^+a^+x^y^  —  y^  +  y^. 

108.  3a;3_|.^2(2a-9)  +  :?^(3-6a)+2a. 

109.  a;*  —  a;3  —  :i:2  ^  3  ^  _  2. 

110.  (a+hy-c\a+by-c(a+by+(^. 

111.  rz;4-2a;3_2a;2-22:~3. 

112.  l  +  b^+c^-3bc. 

113.  a4+2a2  +  l-5(a2  +  l)+6. 

114.  a;*  -  2  a;2  -  5  ^  +  2. 

115.  (x-yy-x  +  y. 


CHAPTER  VII 

HIGHEST   COMMON  FACTORS.     LOWEST  COMMON  MULTIPLES 
THE   HIGHEST   COMMON  FACTOR 

98.  A  common  factor  of  two  or  more  algebraic  expres- 
sions is  an  exact  divisor  of  each  of  the  expressions.  Two 
expressions  are  said  to  be  prime  to  each  other  when  they 
have  no  common  factor  other  than  1.  The  highest  com- 
mon factor  of  two  or  more  algebraic  expressions  is  the 
product  of  all  the  common  prime  factors.  Thus,  a^  and  2 
are  common  factors  of  '^c^x  and  6  a^J,  and  the  highest 
common  factor  is  2  c?. 

99.  The  highest  common  factor  —  abbreviated  H.  C.  F. 
—  of  several  monomials  is  readily  found  by  inspection. 
Thus,  find  the  H.  C.  F.  of  6  T?y\  12  x^y\  40  x^y\ 

6a^z/2  =  2  .  3  .  a;3  -2/2, 
12  0^^  =  2  .2  .3  .aj2.2^3^ 
40a;y  =  2  .2.  2  .5  .0^.?/', 
H.C.F.  =2  .0^  .2/'=2,t2|/2. 

Note.  The  H.  C.F.  of  two  algebraic  expressions  has  reference  to 
the  degree  of  the  factor ;  the  greatest  common  divisor  of  two  arith- 
metic quantities  has  reference  to  value.  The  H.  C.  F.  of  a  and  a^  is  a  r 
if  a  is  any  common  fraction,  and  equal,  say,  to  |,  the  greatest  common 
divisor  of  a  =  J  and  a^  =  ^j  is  ^j.  The  terms  H.  C.  F.  and  G.  C.  D.  are 
not,  therefore,  interchangeable. 

100 


Ch.  VII,  §100]        HIGHEST   COMMON  FACTORS  101 

THE   H.C.F.    BY  FACTORING 
100.   lo  Find  the  H.  C.  F.  of  a^+ah,  a?+hK 
d^-\-a'b  =  a{a-\-h), 

H.C.F.  =  a  +  6. 

2.  Find  the  H.  C.  F.  of  d^  -  h\  a?  -  h\  a*  _ 

a^-b'=  (d  +  W)  (a  +  b)(a-  b\ 
H.C.F.  =a-6. 

3.  Find  the   H.  C.  F.  of  nfi  -  27,  m^  -  6  m -}-9,  nfi +  m 

-12. 

m^-2T=  (m  -  3)  (m^  +  3  m  +  9), 

m^  —  6  m  +  9  =  (m  —  3)  (m  —  3), 
m^  +  m  — 12  =  (m  +  4)  (m  —  3), 
H.C.F.  =  (m-3). 

4.  Find  the  H.  C.  F.  of  4  ^^  -  7  x^  +  3  ^\x^ -  x^t/  -xf 
^-  2/^  x^  —  2  x?y^  +  i/, 

4  x^  -  7  a^2/'  +  3  ^'  =  (4  a;2  _  3  y")  ix"-  -  f) , 

x?-o(?y^  xf-  +  f  =  o?{x-y)-  2/2  (x  —  y), 

=  (^^  -  2/')  (^  -  2/)  =  (^  +  y)  (^  -  2/)  C^-2/). 
oc^  —  2  x^y^  +  .V*  =  (0^2  _  ^2)2  ^  (^  ^  y^  Q^  _  2/)  (aj  +  2/)  (^  -  2/) j 
Il.C.Y,  =  (x  +  y)(x-y), 

The  H.  C.  F.  of  several  algebraic  expressions  is  found  hi/ 

taking  the  product  of  the  common  prime  factors   the   least 
number  of  times  they  occur  in  any  of  the  given  expressions. 


102  ELEMENTARY   ALGEBRA  [Ch.  VII,  §  100 

EXERCISE   XL VIII 

Find,  by  factoring,  the  H.  C.  F.  of  the  following  ex- 
pressions : 

1.  ab  +  a^  b^+b.  4.    am—an  +  bm  —  bn^am—an, 

2.  15  2;  -  9,  6  -  10  X.  5.    (x  +  1)2,  ^2  ^  ^^ 

3.  ax^  —  2  axy^  2  ax^  —  axy.   6.    a2  —  4  a  +  4,  3  a6  —  6  6. 

7.  a2  -  6  a  +  9,  a6  -  3  a  -  3  6  +  9. 

8.  a;2  —  1,  ^2  —  07. 

9.  x/^-y\  x^  +  y^. 

10.  a^  +  l,  ^2  —  1. 

11.  25^2-9(5-1)2,  6J-10a-6. 

12.  a;2  +  3a;+2,  a;2  +  a:— 2. 

13.  a;2__9^  +  20,  2)2-16. 

14.  4a2-5a&-6J2,  8a2  +  2a5-3J2. 

15.  x^  +  x^y'^  +  y^,  afi  +  y^. 

16.  ri;2_8a:  +  12,  a:2_7^  +  6^  ^_216. 

17.  3(x  -  1)3,  a:*  -  1,  a;3  _^  ^2  __  2. 

18.  a^  —  2/^,  a;^  +  a;2^2  +  ^4^  2)^  +  4  r?;2^  +  4  a:j/2  -f.  3  ^/S. 

19.  2:r2  +  172:  +  21,   82)3  +  27,   2ri:2+5a;+3. 

20.  m2 -  4 m  +  3,  m2  -  6m  +  9,  m^ -  9m2  +  27m  -  27. 

21.  a^-16,  a3+2a2+4a+8,  3  a^- 2  a2+ 12  a~  8. 

22.  nfi— 9^^,  m^—  3  m2^2_j,  3  mn^—n^^  m^—  mn^+m^n—ri^. 

23.  a;3_27,  2)4+92:2+81,  x3+22;2+6a:-9. 

24.  Safi+2x^-2x-l,  2)4-1,  ^2;+a-J2)— J. 

25.  (a- 6)3+1,  (a-_j)2_l^  (^__j)2_2((x-J>-3. 


]i.  VII,  §101]        HIGHEST  COMMON  FACTORS  103 

THE   H.  C.  F.  PARTLY   BY  FACTORING 

101.  If  difficulty  be  met  in  factoring  one  of  the  expres- 
luus,  the  factors  of  this  expression  may  often  be  found 
y  trial  of  the  factors  of  the  other  expressions. 

Find  the  H.  C.  F.  of 
1-  -  h^\  2  «6  -  3  a56  _  a*62  +  3  ^353  +  ^254  -Za¥'  +  6«. 

By  division,  a-  +  b^  is  not  a  factor  of  2  a"  —  3  a%  —  aV  + 
;  (I.V  +  a^5*  —  3a¥  +  b^;  and  a*  —  d^W  +  ¥  is  a  factor,  pro- 
lucing  the  quotient  2a^  — 3a6  +  6l 

!>  a«  -  3  w'b  -  a'b^  +  3  a%^  +  a}¥  -  3  at'  +  &" 

=  (a*  -  a'b'-  +  b*)  (2a'-3ab  +  b^, 

=  (a*  -  d'W  +  b*)  (2a-b)(a-  b), 

'     H.C.F.  =  (a*-a262  +  6«)(a-6). 

EXERCISE  XLIX 

Find  the  H.  C.  F.  of  the  following  expressions : 

1.  a^  +  5a;  +  4,  3?-x^-^x-l. 

2.  2:3  -  2  a;2  4- 1,  x*  -  1. 

3.  2a;2  +  24a;4-70,  x*  +  7a^-2;2_6a;  +  7. 

4.  a*-2a3  +  4a2  +  6a-21,  3a*-lla2  +  6. 

5.  l  +  2x^  +  23^  +  2x^  +  A  1+a^. 

6.  3^—2 x^t/  +  2 xy^  —  y^-,  ax  +  ix  —  ay  —  hy. 

7.  a!^j^y'i-2xy  +  z{-2-x')-y(2-z)+2x,a?-  4+y^ 

-2xy. 

8.  a^+a^^+b\  a\2m-Sn^+m(2b^-2ab}  +  UnCa-b^. 


104  ELEMENTARY  ALGEBRA  [Ch.  VII,  §§  102,  103 

THE   H.  C.F.  BY  DIVISION 

102.  *  If  the  expressions  are  such  that  they  are  not  readily 
factorable,  the  H.  C.  F.  can  be  found  by  a  process  which 
depends  upon  the  following  principles  : 

1.  A  factor  of  an  expression  is  also  a  factor  of  any  mul- 
tiple of  that  expression. 

Let  a  be  contained  h  times  in  R.  Then  B=ah.  Let 
mR  be  any  multiple  of  R.  Then  mR  =  mah ;  that  is,  a  is 
a  factor  of  mR. 

2.  A  factor  of  two  expressions  is  a  factor  of  the  sum^  or 
of  the  difference^  of  any  two  multiples  of  these  expressions. 

Let  a  be  contained  b  times  in  R  and  c  times  in  S.  Then 
Rz=ah  and  S=ac;  or,  applying  the  preceding  principle, 
mR  =  mah  and  nS=  nac.  Adding  or  subtracting  the  two 
last  equations, 

mR  ±  nS  =  mab  ±  nac  =  a(^mb  ±  ne^  ; 

that  is,  a  is  a  factor  of  mR  ±  nS. 

103.*  Let  A  and  B  be  any  two  expressions,  arranged  in 
descending  order  of  the  same  letter.  Let  A  be  contained 
m  times  in  B^  with  a  remainder  of  0 ;  let  C  be  contained  n 
times  in  A^  with  a  remainder  of  D;  let  J)  be  contained 
exactly  p  times  in  0, 

A)B(m 
mA 
~0)A(n 

TlC 

D)Oip 
pD 


eH.VlI,§103]        HIGHEST  COMMON  FACTORS  105 

Since  D  is  contained  p  times  in  (7,  pD  =  C ;  since  the 
dividend  equals  the  product  of  the  quotient  and  divisor, 
plus  the  remainder,  and  since  0  is  contained  n  times  in  A^ 
with  a  remainder  B^  A  =  nG  +  D\  since  A  is  contained  m 
times  in  B^  with  a  remainder  C^  B  ^  mA  +  0.     That  is, 

C^pB,  (1) 

A  =  nO+B,  (2) 

B  =  mA  +  0.  (3) 

D  may  be  shown  to  be  a  factor  of  each  of  the  equations 
(1),  (2),  and  (3). 

B  has  already  been  shown  to  be  a  factor  of  (7,  since  it  is 
contained  p  times  in  0. 

Substitute  the  value  of  0  from  (1)  in  (2), 

A  =  npB  +  B  (4) 

=  BCnp  +  1)  (5) 

Substitute  the  value  of  A  from  (5)  in  (3);  and  the 
value  of  0  from  (1)  in  (3)  also, 

B  =  mB(np  +  l)+pB  (6) 

=  B(mnp  +  m+p}.  (7) 

Hence,  B  is  a  common  factor  of  A,  B^  and  C, 
Moreover,  B  is  the  Jiighest  common  factor  of  A  and  B. 

From  (3),  B-mA  =  (7,  (8) 

from  (2),  A-nO=B.  (9) 

By  §  102,  2,  a  factor  of  A  and  5  is  a  factor  of  B  —  mA^ 
or  of  (7;  and  a  factor  of  A  and  C'  is  a  factor  oi  A  —  nC^  or 
of  B.  That  is,  a  factor  of  A  and  B  is  also  a  factor  of  B, 
Since  there  can  be  no  factor  of  B  of  higher  degree  than  B 
itself,  B  is  the  highest  common  factor  of  A  and  B. 


106  ELEMENTAllY   ALGEBRA       [Cn.  VIl,  §§  104, 105 

104.*  From  §  103  is  derived  the  statement  of  the  Rule 
for  finding  the  H.  C.  F.  by  division :  arrange  the  expressions 
171  the  descending  powers  of  the  common  letter ;  remove  a 
monomial  factor^  if  any^  from  either  expression^  and  if  the 
monomial  factors  so  removed  have  a  common  factor  write  such 
a  factor  as  a  factor  of  the  H,  (7.  F,  subsequently  found  ;  divide 
the  expression  of  higher  degree  by  the  remaining  expression 
until  the  remainder  is  of  less  degree  than  the  divisor ;  con- 
tinue the  division  with  the  remainder  as  a  divisor^  ajid  the 
former  divisor  as  a  dividend^  as  before  ;  the  last  divisor  ivill 
be  the  H,  O,  F,  if  there  is  no  common  monomial  factor ;  but 
if  there  is  a  common  monomial  factor^  the  H,  0,  F,  is  found 
by  multiplying  the  last  divisor  by  that  factor. 

105.*  The  H.  C.  F.  of  two  expressions  remains  un- 
changed if  either  of  the  expressions  be  multiplied  or 
divided  by  a  quantity  which  is  not  common  to  both  ex- 
pressions, since,  by  definition,  the  H.  C.  F.  is  the  product 
of  all  the  common  prime  factors.  Thus,  at  any  stage  in 
finding  the  H.  C.  F.  by  division,  a  factor  not  common  to 
both  expressions  may  be  removed  by  division ;  or,  if  at 
any  stage  the  expressions  are  such  that  the  first  terms  are 
not  exactly  divisible,  they  can  be  made  so  by  multiplying 
either  of  the  expressions  by  a  quantity  which  will  make 
them  divisible  —  thus  avoiding  the  use  of  fractions  — 
without  altering  the  value  of  the  H.  C.  F. 

1,    Find  the  U.  C,F,  oi  x^ +  2x^-2x^  +  4x^ +  Sx  and 

x^+2x^-2x'+4.x^+3x=x{x^-i-2  0:^-2  x^+4.x+S), 
x^+2x^-x''+8x^+5:x?-3x=x{x'+2x''-x'+Sx-+5x  '[]). 


Cn.Vn,§106]  HIGHEST  COMMON  FACTORS 


107 


The  factor  x  is  common  to  both  expressions ;  therefore  cc  is  a 
part  of  the  H.  C.  F. 

x^  +  2x^-2x-  +  4.x  +  S     a^  +  2x*-    x^  +  Sx'  +  Sx-S 

x^  +  2x^-2a^  +  4:X^  +  Sx 

af  +  4:X^  +  2x-S 

The  remainder  is  now  of  lower  degree  than  the  divisor. 


a^  +  4:i^  +  2x-3 


x^  +  2a^-2x^+   4.X  +  S 
x'^  +  4:a?  +  2x'-   3x 


^2a?-^x'-\-   lx  +  3 

4a;2-fllaj-3 
The  remainder  is  now  of  lower  degree  than  the  divisor. 


x-2 


4.x^  +  llx-3 


x^  +  4.x'  +  2x-3 

4 

4ar^  +  16i»2+    8a?-12 
4a;^  +  lla^^-    3a; 

5a;^  +  lla;-12 

_4 

20aj2  +  44a;-48 
20x'-ir55x-16 


-11 


lla?-33 
x+  3 


To  avoid  fractions,  multiply  the  expression  aj'+4a^+2a;— 3 
by  4.  This  will  not  alter  the  H.  C.  F.,  because  4  is  not  a  fac- 
tor of  4  a;^  + 11  a;  —  3.  Multiply  the  remainder,  5  a.*^  + 11  i»  — 12, 
by  4  to  make  the  expressions  exactly  divisible.  Divide  the 
remainder  —11a;  — 33  by  —11.  This  will  not  alter  the 
H.  C  F.,  since  — 11  is  not  a  factor  of  4  a;^  + 11  a;  —  3. 

x  +  3)4.x'  +  llx-3(^4.x-l 

4:X^  +  12X 

—  a5  — 3 
-a;-3 


Therefore,  the  H.  C.  F.  =  a;(a;  +  3). 


m 


PAAmmrAnt  ALGt^.^nA 


[Cn.VIT,  §105 


2.    Find  the  H.  C.  F.  of  \x^ -It^ -Vlx^ -\^x-\^ 

and  3  :r^  4-  3  r?^^  +  9  a;2  +  9  a:  +  12. 

3  x^  +  3  aj^  +  9  a;2  +  9  aj  +  12  =  3(i^  +  ^^4  _^  3  a;^  +  3  .^  +  4). 
The  monomials  removed  contain  no  common  factor. 


2 

a;4  +  3a;2  +  3a;  +  4 

2a^+2a!^  +  6a^  +  6a;  +  8 
2a^-    a;^-    6ar'-    Sa;^-   5a! 

3a,4+    6ar''  +  14a;2  +  lla; 

2 

+  8 

6  a;^  +  12  ar*  +  28  a;2  + 22  a; 
6a;*-   3«3_i8x2-24a! 

+  16 
-15 

15ar'  +  46a;^  +  46a! 

+  31 

15aj3+4Ga;2+46  0^+31 


^4-a?  +  l 


2a.^4_^3_0^,2_g^_5 


30i»^-  15i»^^-  90a;2-120a;-75 

30aj^4-  92  ar^+  92  a^^.^,  52  a; 


-107ar^-182a^-^-182aj-75 
-15 


1605  x"  +  2730  aj2  +2730  aj+ 1125 
1605  ar"  +  4922  0?  +4922  aj+3317 


-21921-2192  a?2-2192aj-2192 


15a^  +  46aj2  +  46a;  +  31 

15a^  +  15a;-  +  15a; 

31a!2  +  31a?  +  31 
31ar^  +  31a:  +  31 


x^-\-  x+\ 

15aj  +  31 


107 


Therefore,  the  Yi,Q,Y,  ^x" +  x-\-l. 


Ch.  VII,  §105]        HIGHEST   COMMON  FACTORS 


109 


3.    Find  the  U.CF.  oiia^  -  2a^-2x^ -^Sa^-^^x^-ex 
and  i  x^  —  4  x^  +  x^  —  x'^  —  6  X  —  9. 

Ax^-2x^-2x^+8x^-2x'-6x=2x(2a^-x^-a:P+4.x'-x-3). 

23(^--x'-x'+4.x^-'X-3 


4:X^—4:xP-\-   x^—   x^—6x  —9 
4.x^-2af-2x''  +  Sa?-2a^-6x 


-2a^+3x^-Sx^-{-    x'-d 
—  2a^+    a;^+   i^—4:X^+x+S 


2x'-9x'  +  5a^-x-12 


2x'-9x'+5x^-x-12 

2x'-    x'-      a^+  4.x^-      x-3 
2a.^_9a)H-  ^^x^-      x'-12x 


2a;-l 


X  +  4: 


8 a;^-36 0^4-200^-  4a;~48 
15 1 30  ar^- 15  ^^+15  a;  4-45 


2af'_a;2+a;+3 


2aj^- 
2a;^ 


2ar^-      a;2_^ 
-9ar^+5a;2-    a;-12 


x+  3 
a;— 4 


-8a^+4aj2_4a;_i2 
-8aj3  4-4i»'-4i»-12 


Therefore,  the  H.  C.  F.  =  2  oj^  -  i^^  +  a;  +  3. 
A  more  compact  arrangement  of  the  above  example  is  the 
following : 


2x5-   X*-     a:^^.   4^2_     aj_3 


2x5-9a:t+  5x'*^ 


-12x 


8x4-  6x3+  5x2+llx-3 
8x*-80x3  +  20x2-  4x-48 


15  1 80x3-15x2  + 15X+45 


2x3—     x2+      x+  3 


4x6-4x5+   x*-  x2-6x  -0 
4x6-2x5-2x4  +  8x3-2x2-0x 


-2x5+3x4-8x3+   x2-9 
-2x5+   x4+    x3-4x2+   x+3 


2x4-9x3  +  5x2-   X- 

2x4-   x3+   x2+3x 


12 


-8x3+4x2-4x-12 
-8x3+4x2-4x-12 


2x-] 

x+4 
X-  4 


The  H.  C.  F.  of  three  or  more  expressions  is  found  by 
division  by  first  finding  the  H.  C.  F.  of  the  first  two 
expressions;  and  then  finding  the  H.  C.  F.  of  that  result 
and  the  next  expression. 


110  ELEMENTARY  ALGEBRA  [Ch.  VII,  §105 

EXERCISE  L* 

Find  the  H.  C.  F.  of  the  following  expressions : 

1.  4a?  i-Sx-lO,  4x^+1  x^-3x-15. 

2.  3^  +  2a?  +  2x  +  l,3^-2x^-2x-B. 

3.  4a^-6a?-4:x  +  6,123^-2x'^-20x-6. 

4.  6a? +  1  x^- 5 X,  15 a^  +  Bl3^  + 10 a?. 

5.  2x*  +  a?-9a?  +  8x-2,2a^-7a?  +  lla?-8x  +  2. 

6.  4a^  +  na?+4x-S,2a^-Bx?  +  2a?-2x-S. 

7.  8a?  +  2a^  +  a?,  8x^  +  2a?- Sx^  +  2x-l. 

8.  2a^-5a;2-22a:-15,  6x*-21a?-41  a?-14x-80. 

9.  4a?+14ai^  +  20a?+7Qa?, 

8x''  +  2Safi- 8a? -12x^  +  563?. 

10.  2a?-nx^-9,  4 a^  +  11  a;*  +  81. 

11.  x*  +  23?  +  9,  x*-4a?  +  10x?-12x+9. 

12.  6x*-5a?-10x'^  +  Bx-10,  4a? -4x^-9x  + 5. 

13.  6a^-lSa?  +  Sx^  +  2x,  6a:*- 9x3+ 15^^- 27  a;- 9. 

14.  3a;*-a;3_2a-2  +  2a;-8,  6ar5  + 13a^  + 3  a;  +  20. 

15.  9a?-7a?  +  8x^  +  2x-4,    6x^-7 a? -10x^  + 5x  +  2. 

16.  6a?-2x^-lla?  +  5x^-10x, 
9a?  +  Si*-lla?  +  9a?-10x. 

17.  x'^  +  Ba?  +  Sx^  +  9a?-4x^-12x,a?  +  Sa?-x?-3x'^. 

18.  2ai^  +  a?-8x'^-x  +  6,  4x'^  +  12a? -a?-27 x-^ 

19.  6a;6-9a;*  +  lla:3^.6  2:2_i0a,, 
4a:5  +  10  a^  +  10a;3  + 4a;2  +  60a;. 

20.  4a?  +  a?-nx^+9x-9, 
2a?+3x*  +  la?-Qx^-9x-27. 

21.  43^-6a?  +  9a?-5x  +  8,  8a?  +  8a?  +  9. 


Ch.vii,§§io6,io7]  lowest,  common  multiples  111 

the  lowest  common  multiple 

i06.  A  multiple  of  an  algebraic  expression  is  an  ex- 
pression which  contains  all  the  prime  factors  of  the  first 
expression  and  is  therefore  exactly  divisible  by  it.  A 
common  multiple  of  two  or  more  algebraic  expressions  is 
an  expression  which  contains  all  the  prime  factors  of  each 
expression.  The  lowest  common  multiple  of  two  or  more 
expressions  is  that  expression  which  contains,  only,  all 
the  prime  factors  of  each  of  the  given  expressions. 

Thus,  2  a^b  is  a  multiple  of  2  a6  ;  6  a^x^  is  a  common  multiple 
of  2  a  and  3x^i  6ax^  is  the  lowest  common  multiple  of  2  a 
and  3  x^. 

107.  The  lowest  common  multiple — abbreviated  L.  C.  M. 
—  of  several  monomials  is  readily  found  by  inspection. 

1.  Find  the  L.  C.  M.  of  4  a^,  6  a%  12  b^ 

4:a^b  =  2'2'a''b, 
6  a%  =  2  .  3  .  a^ .  &, 
12b'  =  2.2.3'b% 
L.  C.  M.  =  2  .  2  .  3  .  a'  •  b^  =  12a''b^ 

2.  Find  the  L.  C.  M.  of  8  x^i/,  10  xY,  15  a^f. 

8  x'y  =  2  .  2  .  2  .  a;2 . 2/, 
10:x^7/  =  2  '5'X*>  y\ 
15  x^y^  =  3  '  5  -  x^  -  y^y 
L.C.M.  =  2  .  2  .  2  .  3  .  5  .  a;* .  2/'  =  120  xy. 

Note.  As  in  the  case  of  the  H.  C.  F.  there  are  two  forms  of  the 
L.  C.  M.,  one  being  the  negative  of  the  other. 


112  ELEMENTARY    ALGEBRA  [Ch.  VII,  §  108 

THE   L.  CcM.    BY  FACTORING 

108.    1.    FindihGL.C.M.  oi8(^a^-P),4:a^  +  8ab  +  4:b% 
a^^2ab  +  b\ 

3(a2-62)=3(a  +  6)(a-6), 

4  a-  4-8  a&  +  4  6»  =  2  .  2(a  +  &)(a  +  6), 

a^_  2  ab  +  W=(a-h)(a-  h), 

L.C.M.  =  2  .2.3.  (a  +  6)(a  +  6)(a-&)(a-5), 

=  12  (a' -by. 

2.  Find  the  L.  C.  M.  of  x^- 8a; +  15,  a;^- 3  a;- 10, 
x^  —  X  —  6,  a^  —  6  x^  —  X  +  SO. 

ix^-Sx  +  15=(x-3)(x-5), 

a;2-_3a;_10=(a;  +  2)(a;-5), 

x''-x-6  =  (x  +  2)(x-3), 

a?-.ex^-x  +  30  =  (x  +  2)(x-3){x-5), 

'L.C.M.=  (x  +  2){x-3)(x-5). 

3.  Find  the  L.  C.  M.  of  x^  +  xY  +  2/^  ^  +  y\  x^-f. 

a;4  +  r^y'^  +  /  =  (a;-  +  xy  +  if)  {x^~xy  +  y'^), 

^-\-y^={^  +  y)  C^*'  -  ^y  +  y% 

^  —  f=^(x  —  y)  (aj2  +  xy  +  t/^)  , 

L.  C.  M.  =  (a;  +  2/)  (^  —  y)  ('^"  +  i»2/  +  2/^  (^  -  «^2/  +  2/^- 

Rule  :  Separate  each  expression  into  its  prime  factors^ 
and  write  the  product  of  all  the  different  prime  factors,  giv- 
ing to  each  priyne  factor  the  highest  exponent  which  it  has  in 
any  of  the  given  expressions. 


Ch.  VII,  §  108]  LOWEST   COMMON   MULTIPLES  113 

EXERCISE  LI 

Find,  by  factoring,  the  L.  C.  M.  of  the  following  ex- 
pressions : 

1.  2  :^y^,  3  o?y\  5  x'^y,  7  xy"^. 

2.  4  Q(?y^  5  2;2z/3,  6  xy^^  15  y^, 

3.  4  a%,  6  a^J^  18  a^J^,  36  a%^, 

4.  5  a^JV,  7  ^2^7^^  91  ^453^^  65  d}¥(f^. 

6.  (a  +  S),  (a2  -  52),  ^2  +  2  a6  +  52. 

7.  a;2  —  2/2,  o;^  +  2/^  ^^  —  y^* 

8.  a*  -  h\  a^  +  2  aW  +  5*,  a^  +  5^. 

9.  (m  —  n)^,  m^  —  9^i-^,  n&  —  7i*. 

10.  o;^  —  2/^9  ^^  +  2/^  ^^  ""  2/^* 

11.  m2  —  2  m  —  3,  m^  —  27. 

12.  ofi  —  Ixy  -\-  2/2  —  1,  (2;  —  2/)^  —  !• 

13.  :i:3  _j_  64^  ^2  +  ^  _  12. 

14.  ^12  —  2/12,  ^6  _|_  y<o^  ^4  ^  2/4. 

15.  4  a;2  +  4  a;  +  4,  6  x^  -  6. 

16.  3  (a -5)3,  27^3-27  53. 

17.  3  :c2  ^.  14  ^  +  8,  27  :?;3  +  8. 

18.  (a  -  5)2  -  (?2^  (a  -  5  +  (?)2. 

19.  a;3-5a;+2,  2;4-16. 

20.  2;3  +  2:  -  2,  (a;2  +  2)2  -  x\ 

21.  (a;  —  2/)^  —  ^^  G^  —  2J)2  —2/2,  (y  —  zy^—a?. 

22.  6  ic3(^2;3  —  2/3),  3  0^2/ (a;  —  2/)^  2  2/^ (x^  —  2/^). 


114  ELEMENTAUy    ALGEBRA  [Ch.  VU,  §  109 

THE   L.  C.  M.    OF   EXPRESSIONS  NOT   READILY   FACTORABLE 

109.*  If  the  expressions  are  not  readily  factorable,  the 
H.  C.  F.  can  be  found  by  §  104  ;  and  the  expressions  then 
split  into  factors  by  dividing  each  of  them  by  the  H.  C.  F. 

1.  Find  the  L.  C.  M.  of  4  a;*  +  12  a^  +  2  2)2  _  g  a;  ^.  2 
and  6x^  +  Za^-39x^-21x  +  15. 

The  H.  C.  F.  is  found  by  division  to  be  a^  +  2  a;  -  1.  Tlje 
factors  of  each  expression  are  now  found  by  dividing  each 
expression  by  tlie  H.  C.  F. 

4a;<  +  12af'+   2x'-  Sx+   2  =  2(a!^+ 2a;- l)(2a;2  +  2x-l), 

6x*+   3ar''-39a;2-21a;  +  15  =  3(a-2  +  2a;-l)(2a-2-3a;-5), 

=  3  (a;^  +  2  a;  -  l)(a;  + 1)  (2  a;  -  5), 

L.  C.  M.  =  6  (a-2  +  2  a- -  l)(a- +  1) (2  ic  -  5)(2  a-2  +  2  a;  -  1). 

EXERCISE  lill* 

Find  the  H.  C.  F.   and  the  L.  C.  M.  of  the  followini 
expressions : 

1.  Qa^  +  1  x^-5x,15x*  +  31a^-]-10x^ 

2.  12  a;2  ._  29  a;  +  14;  18  a;2  +  3  a.  _  10. 

3.  9a;*-a;2  +  10a;-25,  6a;*-2a;3+72;2  +  3._5^ 

4.  Qa^-r^-Ux'^-x  +  6,2a^-3x^  +  2x'^  +  x-6. 

5.  12a;3_2a;2_20a;-6,  4a:3_ex2-42;  +  6. 

6.  4a:3-12a;-8,  a;*-6a^-8a:-3. 

7.  a^  +  -ix^  +  ix  +  8,x^  +  Sx'^  +  4:x  +  12. 

8.  2a^  +  a;  +  4,  3a;3  +  iiaj_2a;*  +  12  +  4a?. 

9.  a^-1,  a:3  +  a?-3a;+l,  a;*  +  3a;3  +  ^_l. 

10.    3^Jrx-6,a^  +  2a?-10x-21,afi-8  3^-5x  +  U. 
ii.   3^  +  53^ +  6x  +  8,a^  +  23fi  + x'^~4:,a^  + 5 x^  +  2x-S; 


CHAPTER  VIII 
FRACTIONS 

110.  The  indicated  quotient  of  two  expressions  a  and  6, 
written  in  the  form   -,  is  called   an   algebraic   fraction. 

0 

Since  a-^b  also  expresses  the  quotient  of  a  divided  by  6, 
the  forms  --  and  a-^-b  are  equivalent.     Hence,  by  §  26, 

b 

As  in  arithmetic,  the  dividend  is  called  the  numerator, 

and  the  divisor,  the  denominator,  of  the  fraction.      The 

numerator  and  denominator  are  called  the  terms  of  the 

fraction. 

111.  If  the  terms  of  a  fraction  are  both  divided^  or  both 
multiplied,  by  the  same  quantity^  the  value  of  the  fraction 
remains  unchanged. 

Let  -  be  any  fraction,  and  let  the  quotient  be  q.      By 

definition, 


J  =  ^' 

by  Ax.  3, 

a  =  bq^ 

by  Ax.  3, 

am  =  bmq^ 

by  Ax.  4, 

am            a 

or. 

a      am 
b      bm* 

116 

116  ELEMENTARY   ALGEBRA  [Cii.  VIII,  §  11: 

REDUCTION   OF  FRACTIONS   TO   LOWEST  TERMS 

112.  An  algebraic  fraction  is  said  to  be  in  its  lowes 
terms  when  the  terms  of  the  fraction  contain  no  commoi 
factor.  |- 

Rule   for   Reduction   of    a    Fraction   to   Lowest   Terms; 

divide  both  numerator  and  denominator  by  their  H,  O,  F, 

1.  Reduce  ^^    ,  f  ^  to  lowest  terms. 

36  a^x^y^ 

3  aVif  _     (3  a^xh/)  (x)     __     x 
36  a'xY  ~  (3  a^xY)  (12  ay)  ~  12  ay" 

2.  Reduce  -- — -—- .,    ^  '^ — - — r to  lowest  termss 

x^—2  x^y  +  z  xY  —  ^  ^y  +  y 

x^  —  y^ (.f^  4-  f)(^  +  y)  (^  —  y)  __^-\-y 

x^-2x^y-\-2xY—2xy'^+y^      (x^  +  y"){x  —  y){x  -  y)      x  —  y 

*3.*   Reduce  ^  ""  '   .^ ^—- — ^  to  lowest  terms. 

x^  —  x^  —  b  x^  —  1  X—  b 

By  §  104,  the  H.C.F.  is  discovered. 

x''-^a^-4.x'-\-l  x-{-b^{x'-2x-b){x'-x-l) 
aj4_;^_6if2-Tic-5        [x-  -2  x-b){x^  -\-x  +  l)' 

__x-  —  x  —  l 

~  x^  +  x  +  1 

The  process  of  dividing  both  numerator  and  denomi- 
nator of  a  fraction  by  the  same  quantity  is  often  called 
cancellation.  Cancellation  can  exist,  therefore,  only  be- 
tween the  factors,  and  never  between  the  terms. 

Thus, i—  is  not  equal  to  —  =  -,  since  no  factor  is  cor^ 

3.T-f  a  ^  ^x     3 

men  between  numerator  and  denominator. 


i  eH.vln,§ii2] 


FRACTIONS 


11? 


EXERCISE  LIII 

Reduce  the  following  fractions  to  lowest  terms : 


13. 


14. 


15. 


16. 


7  a*Jc3 
2  o?V>c 

12a36% 
20  a'^hfi 

352;/ 

2a-2h 

a^-^ah  +  ¥ 

(a  —  b)c  —  (a  —  h)d 

(a  +  b)c  —  {a  +  h')d 

ae  —  ad  +  bc  —  bd 

ac  +  ad  +  be  +  bd 

T^  +  x^  —  x—1 

3(2^-1) 

a2-2a6-362 

a^^4,ab  +  3P 


72  2^3^  V' 
8a +  85 
9a  +  9j^ 

mx  —  my 
mp  +  mq 

c?  -f-  ah 

IS. 
19. 
20. 
21. 
22. 


10. 


11. 


12. 


ax -{-a 
h  +  lx 

52  +  5 
1  +  6* 

x^--Y 

3(2^-1)' 

4(2/ +  1)" 


4a3.,_5«5_662 

8a2  +  2a6-362 

a4  +  a2  +  i 

a3_l 

a2_J2_^+26e 

a2  +  52  _  ^2  +  2  a6 

a6(2;2  +  ?/2)  +  xyifj^ 

+  J2) 

abix^-y^y  +  xyicfl 

-62) 

afi  +  x^--[Sx-4: 

^  +  2x^-16x-5 


By  means  of  §  104,  reduce  the  following  fractions  to 
>vvest  terms  : 


or 


23.^ 


24.^ 


25.^ 


x^-6x^-^16x-15 
0:^-6:1:^  +  12  a;2_9^-10 
2:^4^  5^_5:^2__82:-4* 


m* 


^  —  5  ^^  +  5  m^  +  4  rw^  —  5  m  +  6 


a       ,    — a 

--V 

b~^  b  ~ 

a      ^(—1) 
b     5(-l) 

—  a 
-b 

a      —a 

—  a 

a 

b      -b 

b 

-b 

118  ELEMENTARY  ALGEBRA    [Ch.  VIII,  §§  118  11 

THE   LAWS   OF   SIGNS   IN  FRACTIONS 

113.  Since  a  fraction  is  an  indicated  quotient,  the  laws  o 
signs  are  derived  from  the  laws  of  signs  in  division,  §  26 

Therefore, 
By  §106, 

Hence, 

From  the  foregoing  laws  is  derived  the  Rule  for  Change* 
of  Signs  in  Fractions  :  The  value  of  the  fraction  is  not  changea 
if  (1)  the  signs  of  the  numerator  and  denominator  an 
changed  simultaneously^  or  if  (2)  the  sign  before  tht 
fraction  and  the  sign  of  either  the  numerator  or  the  de- 
nominator are  changed  simultaneously. 

114.  If  the  numerator  and  denominator  are  expressed 
in  factors,  since  by  the  laws  of  signs  in  multiplication, 
§  25,  the  product  of  an  even  number  of  positive  or  nega- 
tive factors  is  positive,  and  the  product  of  an  odd  num- 
ber of  negative  factors  is  negative,  the  value  of  the  fraction 
is  not  changed  if  (1)  the  signs  of  an  even  number  of  factors 
in  the  numerator^  or  in  the  denominator^  or  in  both  of  them., 
are  changed ;  and  if  (2)  the  sign  before  the  fraction  and' 
the  signs  of  an  odd  number  of  factors  in  the  numerator.,  or 
in  the  denominator.,  or  in  both  of  them.,  are  changed, 

1  (^  — a)(6  — g)  a»^  _        (a  —  b)(a  —  b)       __     1 
(a  —  b){a—  b)(a  —  h)      (a  —  b)(a  —  b)(a  —  b)      a  —  b 

2  &  —  a)(c  ~  d)(m  —  n)  _  _  (a  —  b)(c  —  d)(m  —  n)  __  _  ^ 
(a  —  b)(d  —  c)  (n  —  m)  (a  —  b)(c  —  d)  (m  —  n) 


Cm.  VIII,  §114] 


FRACTIONS 


119 


[The  numerator  and  denominator,  or  either  of  them, 
%j  consist  of  several  terms.  A  change  of  sign  of  the 
jmerator  or  denominator  means  a  change  of  the  sign  of 
lery  term  of  the  numerator  or  denominator. 


Thus, 


■x^-\-2xf/—7r  _  a;-  —  2  xy  +  y^ 

—  X-  +  y'  x'  —  7/ 

__  __  x^  —  2  xy  -\-y'^  __  _  —x^+2xy  —  y^ 


-x'-^t 


x^  —  y^ 


EXERCISE   LIV 

Reduce  the  following  fractions  to  lowest  terms 


1. 


5. 


x^- 


6. 


ax 


a52- 


aC" 


(c-b)(c-d) 


m^ 


TV"  —  7W 


(h  —  a)2  —  x^ 

a?  —  ac  —  ah  +  he 
he  —  ah  +  ac  —  a^ 


9. 


10. 


l2-x-x^ 

6a;2+22:-60* 

a?-4:x^  +  x  +  Q 
ofi  —  ^aS^  +  Wx—Q 

{a-h^(h-e^(ie-d^ 
(a  +  h^(h  +  e^{d  —  c) 

lio  +  ^x-^x^-^' 


11. 


12. 


13. 


14. 


{x—  a^(x—  h^(x—  g)(g—  ^) 
(a;+  a^(h  —  x)(^e—  x^(c  —  x^ 

(^a^b)(h-e)(e-dXd-a^^ 
(a  +  h)(e -  h)ld  -  e){d  +  a') 

—  m'^+2  m^n  —  2  mn^  +  n^ 
(n  —  m)  (n  —  rn)  (n  —  m)  (m  —  n) 

(a^-h)  (h^  -  e)  (g2  -  d)  (d^  -  a) 
lb  -  a2)((?  -  h^Xd  -  e^Xd?  +  a^ 


120  J:L^MENTARY  AtG^^BliA    [Cii.Vm.  §§115,1  H 

115.  An  integral  expression  is  one  that  does  not  contaii 
any  literal  quantity  in  the  denominator  of  any  term. 

Thus,  2  a^  +  3  ab'^-\--  is  an  integral  expression. 

A  fractional  expression  is  an  expression  which  contains 
a  literal  quantity  in  the  denominator  of  one  or  more  oi 
its  terms. 

Thus,  x^  +  xy-{-^  is  di>  fractional  expression. 

A  mixed  expression  consists  of  an  integral  expression 
and  a  fraction. 

Thus,  a  +  -  and  oc^  +  xy  +  y^ ~     are  mixed  expressions^ 

b  ^       x^-^2 

If  the  numerator  of  a  fraction  is  of  higher  degree  than 
the  denominator,  the  fraction  is  called  an  improper  fractions 
if  the  numerator  is  of  lower  degree  than  the  denominaton 
the  fraction  is  called  a  proper  fraction. 

Thus,  — -^  is  an  improper  fraction ;  and  "^  is  a  proper 
fraction.      "*"  "^ 


REDUCTION  OF  IMPROPER  FRACTIONS   TO   INTEGRAL  ORt 
MIXED   EXPRESSIONS 

116.  If  the  denominator  is  a  factor  of  the  numerator? 
the  quotient  is  an  integral  expression ;  if  the  denominatoi? 
is  not  a  factor  of  the  numerator,  the  quotient  is  a  mixeci 
expression. 

^3  7.3 

Thus,  =  a^  +  a6  4-  &^  is  an  integral  expression, 

a  — 6 

^3     I     7.3  O  7.3 

and  "^     =  a^  +  a6  +  6^  H is  a  mixed  expression. 

a  — 6  a  — 6 


Ch,  VIII,§I16J 


FRACTIONb 


121 


Rule  for  Reduction  of  an  Improper  Fraction  to  Integral 
or  Mixed  Expression :  Divide  the  numerator  by  the  denumi- 
nator. 

Thus,  ^^±jl±l  =  :,^  +  ^  +  l. 


1.    Reduce    ,^   ^  :^ ■ —  to   an   integral    or 


mixed  expression. 


2x'  +  x-l 


2x^  +  x-l 

4rx'-{-2x^-2x^ 

—  6a^H-5a;^  +  i^  +  l 

^ex'^Sx'+Sx 

Sx'-2x  +  l 
Sx'  +  4.x-4: 


2x'^-Sx  +  4: 


-6x  +  5 

4.x'-Ax^  +  3x''  +  x  +  l^^^      3a;  I  1  I      -^^  +  ^ 

2x^^x-l  "^     "^2x^  +  x-l 


Or,  by  §  113,  =2x'-3x  +  4.- 


6  x  —  5 


2  a;2  ^  a;  - 1 


EXERCISE  LV 


Reduce  the  following  improper  fractions  to  integral  or 
mixed  expressions : 


2. 


X^  —  X1/  +  ^2 

X 

2x^  +  Ax  +  \ 

2x 

aP'  +  ixy-  .«/2 

x+y 

3  to2  +  4  TO  +  b 

s. 


6. 


TO  +  1 


^  —  y^ 
x+y 

2:2+2 

a^-23?  +  2x^  +  x-l_ 

x^  +  x-1 

a^-Sa^  +  2a-l 


122  ELEMENTARY   ALGEBRA  [Ch.  VIII,  §  117 

REDUCTION  OF  FRACTIONS   TO   EQUIVALENT  FRACTIONS 
HAVING   THE   LEAST   COMMON  DENOMINATOR 

117.  As  in  arithmetic,  the  least  common  denominator 
—  abbreviated  L.  C.  D.  —  of  a  number  of  fractions  is  the 
L.  C.  M.  of  the  denominators. 

w      T3    -,  2  m     6  m^     12  mn    ,  •     i     i.   j      ^  • 

1.  Keduce  •— — ,   -:: — ,     _,^^    ^     to   equivalent   fractions 

6a^      b  a       10  a^ 

having  the  least  common  denominator. 

The  L.  C.  M.  of  the  respective  denominators  is,  §  107,  30  a?. 
Take  30  a^  as  the  L.  C.  D. ;  and  divide  30  o?  by  the  respective 

denominators,  3  a?,  5  a,  and  10  a^,  thus  obtaining  the  respective 

quotients,  10  a,  6  a^,  3. 

2  m(10  g)^ 20  am.       ^  mH%  a^)  ^^Q  a^m^ ^      12  mn(3)  ^ 36  mn 

3  a^lO  a)  ~  30  a^  '       5  a{Q  a')         30  a'  '       10  a\3)       30  a^  * 

1  1 

2.  Reduce —    and to 

a"  —  an  —  ac  +  be  oc  +  ac  —  ao  —  e^ 

equivalent  fractions  having  the  L.  C.  D. 

Factor  each  denominator,  and  simplify  by  §  114,  if  possible  : 

1^1 

a^—  ab  —  ac+  be     (a  —  6)(a  —  c)' 


be  +  ac  —  ab  —  c^      {b  —  c)(c  —  a)      (c  —  b)(a  —  c) 

The  L.  C.  M.  of  the  denoininators  is  (a  —  b)(a  —  c)(G  —  b). 
Divide  the  L.  C.  D.  by  the  factors  of  the  respective  denominators, 
thus  obtaining  the  respective  quotients,  c  —  b  and  a  —  b. 


(a  -  b)(a  -  c)(c  -b)      (a  -  b){a  -  c)(c  -  b)' 

1  (a—  b) a  —  b 

(c  —  b)(a  —  c)(a  —  b)      (a  —  b)(a  —  g)(g  —  b)' 


Cii.  VIII,  §117]  FRACTIONS  123 

Rule  for  the  Reduction  of  Fractions  to  Equivalent  Frac- 
tions having  the  L.  C  D. :  Simplify  each  fraction^  and  express 
the  denominator  as  the  product  of  prime  factoids.  Take  the 
L,  C.  M,  of  the  denominators  as  the  L,  O,  D.  Multiply  both 
terms  of  each  fraction  by  the  quotient  found  by  dividing  the 
L,  C,  D,  by  each  denominator, 

EXERCISE  LVI 

Reduce  the  following  fractions  to  equivalent  fractions 
having  the  L.  C.  D.  : 

1.   J_,    A.,    i.  6.   -i-.    ^±^. 


2. 


10  a      4  a^      a^  x  +  y  X  —  y 

20^      5x         6  „    a  —  c  a  +  c 

,  . • 

c—  d  c  +  d 


8y 

2/      12  y^ 

3x- 

y,    ?/-^^. 

la^ 

4:  am 

a—h 

a  +  b 

5xz 

\2x^ 

a 

a 

a  +  b 

a  —  b 

11.    — 

1 

9 

8. 


5.    -•»     -•  10. 


x^  -{-  y      x-\-  y^  ^ 
x^  —  y'^      X  -\-  y 

x^  y'^ 

a?  —  y^     x^  —  y^ 

Sx  —  4:y     5y  —  8x 
x^  —  y^         y^  —  a? 


12. 


13. 


14. 


(a  —  b^(m  —  n)  (J  —  a}(m  +  ?i) 

a^  V^ 

(^a-b^(b-ci  (b-a){c-b^^ 

cP"  —  ac  b^—  be 


(a  -  J)(J  +  c}Qa  -c)      (a  -  6)(J  -  c}(c  -  a) 

2m-3  3m4-7 

2m^-5m-{-o'    3 m^ -  2  m^  -  18 m  -f-  7 * 


124  ELEMENTAllY   ALGEBRA    [Ck,  VIII,  §§  118, 119 

ADDITION  AND   SUBTRACTION  OF  FRACTIONS 

118.  By§56,  ^+^  +  ^  =  %l^  +  g. 

*^  a  a      a      a 

Therefore,  tlie  sum  of  a  number  of  fractions  having  a 
common  denominator  is  a  fraction  whose  numerator  is  the 
algebraic  sum  of  the  numerators  and  whose  denominator  is 
the  common  denominator  of  the  fractions. 

By  the  law  of  signs  in  fractions,  §  113,  a  fraction  in  the 

form  —  --  may  be  changed  to  the  equivalent  form,  -] — — -. 
b  h 

Find  the  alerebraic  sum  of  - — h — 

""  3  a      3  a      3  a  ^ 

By§113,  -A==  +  =:^.  I 

3  a  3a 

The  algebraic  sum  of  the  numerators  =  2  x  +  ^x^  —  h. 

The  common  denominator  =  3  a. 

2x     bx''      -b^2x  +  5x^-b 
3a      3a      3a  3a 

119.  If  the  denominators  are  not  common,  the  L.  C.  D.| 
may  be  found  by  §  117  and  the  fractions  added  as  before. 

1.    Find  the  algebraic  sum  of  ^^  \-- tt' 

3  a      2x     a^x 

The  L.  C.  D.  =  6  a^x. 


2x 

_2ss(2ax) 

iaa^ 

Sa 

3  a  (2  ax) 

6a'x 

3a 

_3a(3a')  _ 

9a^ 

2x 

2a;(3o^ 

6a'x' 

1 

_-l(6)     . 

* 

-6 

a^x 

a'x{&) 

6a^x 

2x  ,  3a  ,   -1     4.ax^-^9a^-6 

J 1 = — ^* 

3  a     2  X      a^x  6  a^x 


Ch.  VIII,  §  120]  FRACTIONS  125 

Rule  for  Addition  (or  Subtraction)  of  Fractions :  Reduce 
the  fractions,  in  their  lowest  terms^  to  equivalent  fractions 
having  the  least  common  denominator ;  the  sum  of  the  frac- 
tions is  a  fraction  tvhose  numerator  is  the  algebraic  sum  of 
the  Jiumerators  and  whose  denominator  is  the  least  common 
denominator  of  the  fractions. 

120.  It  should  be  carefully  noticed  that  the  sign  of 
division  in  fractions  is  a  sign  of  aggregation. 

o?  4-  ob  4-  h^ 

Thus,  '      —  means  that  the  whole  of  the  numerator, 

a^-2  ah  +  Ir 

a^  -\-ah  +  b^,  is  to  be  divided  by  the  whole  of  the  denominator, 

d^  —  2  ah  +  Ir.     If  the  fraction    ^  ^  be  preceded  by  the 

a-  —  2  ab  -{-¥ 

minus  sign,  the  whole  process  is  indicated : 

-  (a^  -\-ab  +  b")  ^(a''-2ab  +  b% 

That  is,  the  minus  sign  before  the  fraction  is  to  be  inter- 
preted as  affecting  the  whole  of  the  numerator. 

Thus, 

a^b a—b      _  (a+-b)  —  (a—b)  ^        2b 

a'+ab  +  b'^     a:'+ab+b^~     a'+ab-[-U'     '^  a^+ab+b^' 

or,  by  §  114, 

a-\-b  a—b  a+b        .      —a-\-b    __        2b 


a^+ab+b^     a^+ab  +  b^     a'+ab  +  b^     a^+ab  +  b^     a'+ab  +  b'' 

1.    Find  the  alsrebraic  sum  of   — : 

The  L.  CD.  =  12 a. 

a-\-b^4:(a  +  b)^        a—b^     S(a  —  b) 
3  a  ~~     12  a     '  4  a  ""         12  a    * 

a-{-b     a-b_4:(a  +  b)~3(a-b)__4a-\-'ib  —  3a+3b_a-\-7b 
3a       4a  ~  12a  ""  12a  *"  12a  ' 


126  ELEMENTARY   ALGEBRA  [Ch.  VIII,  §  120 


EXERCISE  LVII 


Find  the  algebraic  sum  of  the  following  fractions  i 

4.    ^  +  A.  7.    7  +  1. 

OX      4:X  a 


1. 

i_£. 

m     m 

2. 

3  a      a 

7       7 

3. 

m     n  ^ 

a^     X 

1  .  1 


a  —  a; 


5.  _^  +  -i^.  8.   ^^^ — ^  +  1. 
6a      96  X 

6.  4-^-  9.    1-         ^ 


^     xy  a-\-h 

ba-Yll      lh-2a     a-4b 
16        "^       12  8      ■ 

..     2a;-5.y-3      3a;-8.y  +  45  ,  ^ 
"•  15  25  "^"' 

a  +  4  6      2(a-36)      ll(a  +  6) 
■       10  15  20       ' 

13     3(x-y)      5(2x-3.y)      7(x-2,y)_ 
4  6  8 

5^  +  16  6-14  ,  a-66  +  7      3a  +  14J-15 


14 


33(a  +  l)  36(a  +  l)  44(a  +  l) 


4a4-7  6'      3      45-3a      3a-2c 
6  be         c         2  ah  ac 

5a;  +  3y     y  +  2g      3a:  +  4y     x  +  ?,z       1 
?>x^  iyz  ^xy  4xz        4  a; 

^^    a2  +  4  5c      62_3ac     (16  + 4^2       ^    ,    ^  . 
Sac  6a6  46e  3a      2b' 

2       2(6 a- b)       1       a  +  h      Sa^-b^ 
■   5a  15  a6  3  J      10a2        10a26  * 

a-2b      Sa-4b       1       Ga-5b     a^  +  2b'^ 
'      4ab  6a2         2b        20b^  Za% 


Ch,  VIII,  §  120]  FRACTIONS  127 


20. 


4:(ah  +  xy^      ay—^lfi      a^  —  5bx       h y 


15 hx  ^ab  6  ax         2a      3  6      10a; 


21     (1  +  ^)^      (l  +  y)"  I  2a;-2.y  +  l  I  2x+^/ 

-1  -^  --  ^  ^y7 

b 


22.     -  H -•  27. 


x^ 

y' 

2 
3 

+  1 

a  — 

\ 

a 

a  + 

m 

h 

1  + 

m 

X 

_^_ 

X  — 

■J.. 

y 

X 

x  +  y 

xy  xy'^ 


23.    ---^ 28. 


24.    -  — ^  — "T ^.  29. 


3  2 


a;  — 11      a; 

-7 

3(x  +  2) 

cc  +  5 

b(x 

2x 

-2). 
+  1 

x-13 

X 

-18 

25. . 30. 

1-x     1  +  x  10(a;-3)      14(a;-2) 

26.    -1^--U.  31.  '^  3 


a-6      a  +  b  Zx-S      2x-2, 

2.t;-13        3a;-16 


O^. 

10a;  +  10      15a;  +  45 

33. 

3                5                2 

2a;-4      6a;-12      3a;  +  6 

34. 

1  1   ^-y        2    . 

y      x^  +  xy     x  +  y 

35. 

X-  y       X  —  a          y  —  b 

7               '                           .7 

36. 


37. 


xy        ax  -\-bx     ay  +  by 
5         .      3         13a  +  7  6 


4(a  +  6)      a-b     4{a^-b^) 

5  7  x-4 

x  —  2     x  —  S     a^  —  5x  +  6 


,„    21a;  +  13         5x      ,       16a;-3 

OO.      — —— • j- 


12a;  +  24      Sx-d     ix^  +  ix-^ 


1 

128  ELEMENTARY   ALGEBRA  [Ch.  VIII,  §  121 

121.  If  some  of  the  factors  of  the  denominators  are 
alike  except  that  their  terms  are  not  arranged  in  the  same 
order,  they  may  be  made  to  take  the  same  order  by  §  113. 

1.    Find  the  algebraic  sum  of 

1  1.1 


1  1.1 


(a—b)(b—c)      (6— c)(c— a)      (a—c){a—h) 
1  1 


{a  —  b)(b  —  G)      (b  —  c)(G—a)      {c  —  a)(a  —  b) 

9 

_  c  —  a  —  (a  —  b)  —  (b  —  c) 2  (c  —  a) 

(a  —  b)(b  —  c)(c  —  a)         (a  —  b){b  —  c){c  —  a) 

2 


(a-b)(b-c) 


2.    Find  the  algebraic  sum  of 


'  +=j — A^.+  * 


x+l      1  —  x      — 1— a^     1  +  a^ 
11  2,4 


a;  + 1     ic  - 1      —  1  -  a;2     1  +  a;* 

a;-l-a!-l  2  4^-2  2  4_ 

-1)      ar'  +  l      a;^  +  l      a^-1      a^+1      oc* -i- I 


(a;  +  l)(a!-l)      ar'  +  l      a;'  + 

_2a^-2  +  2a^-2         4^-4  4 

x*-l  "^x^  +  l      a;^-l      a;*  +  l' 

-4a;<-4+4a;^-4_   -8    _     8 

a^  — 1  a^  — 1      1  — a;* 


Ch.  VIII,  §  121]  FRACTIONS  '  129 

EXERCISE   LVIII 

Find  the  algebraic  sum  of  the  following  fractions  : 

^     _L_ 1 L- 

x^  —  1      x  +  1      1—x 

7?  1       ,        x^ 


2. 


3. 


x^  —  ]f      y  —  ^      y^  —  ^ 

a     __    2  a  3  db 

a  —  h      a  +  b      IP"  —  c? 


x  —  Zy      Sy  +  x      9  y^  —  3^ 
1  1 


(a;-l)(a;-2)      (2-x)(S-x)   .  (l-a;)(a;-3) 

6.     I I +  — J . 

(a  —  h^(h  —  c)      (b  —  a'){a  —  c}      {c  —  J)  ({?  —  a) 

c  +  a  b  +  c  a  +  b 


(a  —  b^Q)  —  c)      {a  —  b}(^a—  c)      (^c  —  b)(^c  —a) 

8.     ^! + ^! + ^ . 

(«— 6)(a  — c)      (h  —  a)(h  —  c)      (^c  —  a)(c  —  h) 

(a—m')(b  —  m')      {h  —  m)(c—in}      (c—m')(a  —  m') 
(a  —  c)(b  —  e)         (b  —  a)(c  —  a)        (c— 6)(a— 5) 

10.  a  .  h 


{a-b^ia-cXa-d)      (h-a)(b-c')(b-d) 


(c-a)(c-fi)(c-(;)      (c;-a)(cZ-J)((Z-c) 


a      b-\-a      b—a      a— "lb      —lb  — a 
-1^2  1       _^       1        .        \x  32  :?^ 


a  +  2^      2;?;  — a      a2  +  4;i;2      a^  +  16^* 


130  ELEMENTARY   ALGEBRA  [Ch.  VIII,  §  122 

REDUCTION   OF   MIXED  EXPRESSIONS  TO   FRACTIONS 

122.  Since  a-h  1  =  a^  any  integral  expression  may  be 
written  in  the  form  of  a  fraction  having  the  integral 
expression  for  a  numerator  and  1  for  a  denominator. 

Thus,  a  +  6  =  |  +  |  =  ^. 

Hence,  the  Rule  for  Reduction  of  Mixed  Expressions  to 
Fractions :  Write  the  integral  part  of  the  expression  as  a 
fraction  having  the  integral  expression  for  a  numerator  and 
I  for  a  denominator^  and  add  the  fractions, 

1.    Reduce  a  —  h ■^-—-  to  a  fraction. 

a  +  0 

a-\-b  1  a  +  b        a  +  b 


EXERCISE  LIX 

Reduce  the  following  mixed  expressions  to  fractions : 

1.  a  +  b —---  6.    — \-2  —  m- 

a  +  b  3 

2.  m  +  0 -—'  7.    -— — a  +  b* 

m  —  3  a  —  b 

3.  x^  +  xy  +  y'^ •  8.    x  +  y ^• 

X  —  y  x  +  y 

4.  x^  —  xy  +  y ^ 9.    2a  — 6b — ^f — -• 

^      ^       x  +  y  Sa-2b 

5.  a^  +  ab  +  b^-- — -•        10.    m^-n^ iT~"2' 

a^--ab  +  b^  m^  +  n^ 


Cii.  VIII,  §  123]  FRACTIONS  131 


I 


MULTIPLICATION  OF  FRACTIONS 

a  c 

123.    Let   7    and  --  be  any  two  fractions ;  and  let  the 
b  d 


product  of  these  fractions  be  P. 


Then, 

a     c      p 

(1) 

multiplying  (1)  by  6, 

ah     c      rjy 

(2) 

multiplying  (2)  by  d^ 

0       a 

(3) 

simplifying  (3), 
dividing  (4)  by  bd^ 

ac  =  bdP, 

(4) 

ac  _  -p 

(5) 

applying  Ax.  5  to  (1) 

and  (5), 

1 

a     c      ac 
b     d      bd 

(6) 

r    The  product  of  three  fractions  can  be  found  by  multi- 
plying the  product  of  the  first  two  by  the  third,  and  so  on. 

Rule  for  Product  of  Several  Fractions :  The  product  of 
any  number  of  fractions  is  a  fraction  whose  numerator  is  the 
product  of  the  numerators^  and  whose  denominator  is  the 
product  of  the  denominators  of  the  given  fractions. 

1.    Find  the  product  of  — ^  x  —^  x  ^• 

o  y^        8        10 

2o(?    5 xy     ?/^  __  10 x^]f  _ x^y 
32/2' "s"  *  10""240^~24* 


132  ELEMENTARY   ALGEBRA  lCh.  VIII,  §  122 

In  multiplying  several  fractions  the  process  may  be 
simplified  by  cancelling  the  common  factors  before  finding 
the  product. 

2.    Find  the  product  of  ^^  •  ^+/-^t-^  •  ^P^- 

^  —  y^         XT  —  y^  x^  —  y^ 

■y?  +  ]f    3?  +  xy  4-  if     {x  —  yf 

(x-yy 

(x^)(x^J^j:^-r-f){:x  +  y){x,^^{o^A^){x  +  y){x^^ 

_  {x  —  yY  _  fx  —  y"^^ 
~  (x  +  yy~[x  +  y^ 


EXERCISE  LX 

Find  the  product  of : 

3  3^     a^b^  '  ofi  +  y"^           x-^  y 

^*    9/*  20a^3"  ■  a3  +  63*        12a* 

g    4  m^n^/)  ^  2S  a^y^_  ^^  2x  +  U  ^    a^-9 

7  it^y'^z       m^^p  '  a  —  3       3  a;  +  21 

7  g^m^c*     26  a^y%^  (x  -  ,y)*  _    a^- j/* 

13^^7  ■  21  a%*c3'  ■  a^+'i/2   ■(^_^)S 

.     44  a25a^     26  m%  '  12  '^^  +  ^'''          '^  +  "^ 


■     Qbanfi      33  aJa^  a2_i6     ^2_3^^  q 

®"     32  a*5*~  ■   81  d^<f'd?  '  '   a^  +  u'  a?  +  U 

a^-W-       a*  ah  +  Jfi     a'^-h^_ 

«»      ■  a  +  b  '   a^-b^'  a^  +  js' 


Oil.  VIII,  §  123]  FRACTIONS  133 


15. 


16. 


17. 


18. 


19. 


20. 


21. 


22. 


23. 


24. 


25. 


26. 


27. 


X^  +  X     x^  +  x^  +  1 

nfi—lm +  12      m^—6m  +  8 


nfi-Sm  +  2      m2-8m  +  16 

2a^  +  a%-2ab^-b\  a  +  2h 
a^  +  2a%  +  aJ)^-\2}fi'  2a  +  l 

m^  +2m^  +  2m  +  l  ^  nfi-{-2m  +  \ 
m^  + 1  rrfi  —  1 

^  —  y^  ^  x^  +  y^^ 

x"  +  x^y  +  xy^  +  y^     x^  +  y'^ 

x^  —  x  —  Q        a^  +  S  x^  —  4:x 
x'^y  +  xy  —  2  y       a;^  +  a;  —  12 

2x^-5x+S     Sx'^-llx  +  G  ^  9>-2x-x^ 
Zx^-8x  +  ^'  2x^  +  bx-12'  x^-'^x  +  ^' 

W^         )\?>a-b       A       ?>a  +  h) 

2^:2  + 2^2:- 8:1^- 8  a     2^:2-2  t«.r-3  a;  +  3  a      x  +  2 


x^  —  ax  +  2x—2a  x^  +  ax  —  4:x  —  4:a        2x  —  S 

a:3_2a;-l  x^-x^  +  2x^-x-\-l      x  +  1 


x^  +  2x^  +  2x  +  \  ofi-\  aj2  +  l 

^  +  ^2^1      *^2  +  l'  ^_1 

7yi2  — a2_|_2  ^—1         (??2  +  a  +  l)2  m  —  1 

7Y?-\-2am-\-<£^  —  \     m^  —  am  +  a  —  1     m  —  a  +  1 

7722  +  3  m  —  4 


73i2  +  am  +  5m  +  4:  a  +  4: 


134  ELEMENTARY  ALGEBRA    [Ch.  VIII,  §§  124, 125 


DIVISION  OF  FRACTIONS 

124.  The  reciprocal  of  a  fraction  is  a  fraction  formed 
by  interchanging  the  terms  of  the  given  fraction. 

Thus,  the  reciprocal  of  -  is  -:  the  reciprocal  of  a  is  — 
ha  a 

The  product  of  a  fraction  and  its  reciprocal  is  1. 

Thus,  =  —  =  1. 

n    m     nm 

125.  Let  y  and  -  be  any  two  fractions;  and  let 

0'         a 

'     M=<>.  (1) 

Then, by  §26,  f=^-5'  (2) 

multiplying  (2)  by  ^,    f  •  ^  =  ^  •  J  •  ^  =  <?,  (3) 

cue  ct    c 

applying  Ax.  6  to  (1)  and  (3),      ^  ^  ^  =  ?  .  -.  (4) 

babe 

Rule  for  Quotient  of  Two  Fractions :  Multiply  the  divi- 
dend by  the  reciprocal  of  the  divisor, 

1.    Divide  M£_Ti)!   by  ^^^^ -^> 

35(a2  -  62)      ^  28(a  -  h-)\a  +  J) 

12(a!-iy  .         15(1 -a;) 


4  4 

__n(J'-^0--x)    28(a-^)(»-^(^H^^ 

6  5 

_16(l-a;)(a-6) 
25 


Cn.  VIII,  §  125] 


FRACTIONS 


135 


1. 


EXERCISE  LXI 

Find  the  quotient  of  : 
S  ax        5  a 


8 

55  a 
24^' 


12  xy 
15  aV 


36^  _^  24^3^ 
27  a2J3  '    81  a%  * 

40  a^h^c^  _^  35  a%^c\ 
22nfixh^  '  88m^xz^' 

'a^  +  h^ 


^     9yCa-4:b)  _^  6y\4h-a) 
22  a\v{a  +  J)      55  ax'^Qa  —  6) 

6.    — ^(a— 6). 

8.    (la  +  i5)-(|a+|6> 


a2  +  aJ  +  Sn  .  «*  -  ^* 


10. 


11. 


12. 


13. 


14. 


15. 


16. 


17. 


63      (a2 


-68 
62)2 


raM-63 
[a*  +  6* 

U2  +  62 


■]■ 


a  +  6 
a* +  6* 


a*  — 68 

Vja  +  02  _  c2  _^  a2-_(6-c)2"]  (a  +  6  +  g^ . 
[(J  +  c)2_a2  ■  c2-(6-a)2j(a-6-c)2' 

2)3  _  2  ^2  -  2  a;  +  1      a;2  +  a;  +  l"|    y?~\ 

-1    J(^  +  l/ 


a;2  -  2  X  +  1 


2?- 


.y)''-g^     (ir  +  g)2- 


o2j- 


t/2-(a;-g)2_ 


L8x2-2a;-l  '  3a;-l-2a^J"  5a;-l 


2/  +  2)2         (a;-^  +  2)2j    •    a;2_(^+2)2 

+  2a^ 


r- 


m^  —  m  —  6       2  m^  - 


m- 


m^  —  5  m  +  6     6  m^  —  11  m  + 


J^ 


2  m^  —  m  —  3 
3  m^  —  10  w  +  3 


136  ELEMENTAKY  ALGEBKA  [Ch,  VIII,  §  126 

COMPLEX  FRACTIONS 

126.  A  complex  fraction  is  a  fraction  having  one  or  but! 
of  its  terms  in  the  form  of  a  fractional  expression. 

Thus, -;  — -T-,  and -'  are  complex  fractions. 

X  X 

The  process  indicated  is  merely  one  of  division,  —  aftei 
the  numerator  and  denominator  have  been  simplified. 

Hence,  the  Rule  to  Simplify  Complex  Fractions :  Dividi 
the  numerator  hy  the  denominator. 

1.    Simplify 


a  — 

b 

b 
1- 

a 

b 

a- 

-b 

a\       fa  —  h\ 

—br{-r} 


1 

b—a     a—b 


-^.  =  -1. 


If  the  L.  C.  D.  of  the  denominators  of  the  several  frac 
tions  is  easily  found  by  inspection,  it  is  sometimes  prefer 
able  to  simplify  the  complex  fraction  by  multiplying  botj 
numerator  and  denominator  by  that  L.  C.  D. 

-?  bii-'^ 

hj      h  —  a___A 


'{^) " 


Cn.  VIII,  §  127]  FRACTIONS  137 

i27.    A  continued  fraction,  that  is,  a  complex  fraction  in 
the  form  ,  is  simplified  by  beginning  at  the  last 

fractional  expression  and  working  up^ 
1.    Simplify  j . 


2x 


^ 1_  2a?-l 

2x         •  2x 


2x  2{x-l) 

2a;-l  2x-'l 

1 


2(2  g;  - 1)  ""    -  2  ' 
x  —  1  x  —  1 


-2 


Note.    In  §  52  it  was  stated  that  0  can  never  be  taken  as  a  divisor; 
I  hence;;  if  h  =  0,  the  form  y  may  be  considered  impossible.     The  defini- 
tions of  fractions  hitherto  given  must  be  understood  to  exchide  0  as  a 
denominator. 


138  ELKMKNTAUY   ALGEBRA  [Ch.  VIII,  §  127 


EXERCISE  LXII 

Simplify  the 

follow 

kviiig  fractions : 

1+1 

X 

1               » 

h 
a 

3              r 

y 

^*            h 
c 

m      X 

n      y 

2.   ^. 

X      m 

n      y 

5. 


h  '  a  —  h 
h  a  —  h 
a      a  -\-  b 

y 
1  +  ("^-yy 

4x1/ 


1- 

X  — 

3y 

7 

x  +  y 

Bx 

+  y 

-3 

x  +  S 

X- 

X  — 

-y 

.  2 

8. 

X  — 

3 

x  +  i 

1 

1 

7 

9, 

x-n 

[ 

11 

4 

12. 


1-1 

X 


13. 


1- 


1  —2 

2 


3       lOrr 


x-y 


14. 


X      a%^      23  62 
-  9  ~r 


1- 


10, 


1+1 

1      5  6      a62  « 


a       X        X 


,2 


J2 


15. 


,^       a  —  ha  +  b                                      q,1 
11, _ .  t)  -J 


— +——  1-1 

(a  -  hf      (a  +  iy-  X 


Ch.  VIII,  §  127]  FRACTIONS  139 

REVIEW   EXERCISE   LXIII 

1.    Simplify  the  following  fractions: 

5  a:^  +  4  a:  -  1  a^+Sa  +  2 

5.  1 


12a;2  +  24a:-15 


a^+1  a  +  12 


y  x^  —  y^ 

3.  ■ 


1  __  1         X?  ■\'  y^  ofi—^  y^           x-\-y 

y     x  x^  —  y'^      x^  +  ^xy  +  y"^ 

3  o;^  +  6  a;^  —  3  a;  —  6^  xy*^  +  y^      ^  x^y'^  —  x^^ 

x^  +  S  x^  +  2  X  x^  +  xy  +  y^       y^  ^  ^^ 


9. 


10. 


11 


— ! a 

_b a2_52 

£  —  1         cfi  +  b^ 
b 

a^  —  x^         ,  a^x  +  x^ 


a^—2ax  +  x^       a?  —  x? 
x^-^x  +  20     :r2-13a:  +  42 


Q  X  x^  —  5  X 

lo    /"^  I    ^  — ^A^      fi  b'-a\ 

1    ^  _^  fa  +  x  __  g  — a^Y 
a  —  x)     \a  —  X      a  +  xJ 

fx  +  y     o^_+jf\  _^  fx  +  y  ___  ^Mv^Y 
x^  +  y\x  —  y      x^  —  y^J      \x  —  y     x^-^y^J 


14.  (?-- 1-4 


a;?/ 


140  ELEMENTARY   ALGEBRA  [Ch.  VIIL  §  127 

16        3  2:^;-!  ^^      Sx'^+2x-l 

"^+2  +  2 
2m- 3+1 

18.     —^ :j 19. 


20. 


21. 


^4  _  ^4  ^2  _  a;2 


x3  +  a;2- 

X- 

-1 

a2  +  62 
b 

a 

a? 

-J2 

1_1 
b     a 

+  52- 

2:i;2^5r?:  +  2  ,  2  2;2  4.9a;  +  4 
^2  —  4         *  a;  +  4 

1  X 


22^     ^  -  y      a:^^  -  y^    ^  ofi-2xy  +  y'^^ 

X y  xy 

xy  +  y^     x^  +  xy 


/^^_jf4  ^   a;  +  y  \  _^  fo^_+^  _^    x  +  y  \ 
\x^  —  y'^     x^  —  xy)      \x  —  y       xy  —  ^v 

/-J ah         V-| ah  \  ^  a^  — < 

•  V        a^^ah^hV\        a?  +  2ah  +  hV  '  a^-{-. 


a^  —  he  V^  +  ca  c^  +  ah 

25. 7Z~z r — 1 — 71  .   ,  ,  T"  ~r  ' 


?6. 


(a  — 5)(^  — ^)      (5  +  c)(ft--a)      ((?  — ^)((?H-i) 

a;2  —  ^^^  y"^  —  zx  ^  —  xy 

{x  +  ^/)(a:  +  ;2)       (?/  +  5:)(2/  +  :r)       (aJ  +  x^\z  +  j/) 


1  a 

27     5 15 28. 


^2      ^  +  1  1 a_ 


X  +     -"  1 i- 

a;  —  1  a  —  1 


Ch.  VIII,  §  127] 

a  —  h 


FRACTIONS 


5-. 


29. 


\-\-ab      1  +  hc 
.         (^a  —  b)(b  —  e^ 
~  {l  +  ab){l  +  be} 


30. 


^24. 


M 


^2+52 


(^2-52) 


a+b      '2\ 


a+  b 


31.    (a2  +  62). 


J4 


J2- 


a  +  b      a  —  b 
-1 


32. 


33. 


34. 


35. 


36. 


l-\-'l/\l-^X 


1- 


x^+2/'^~x  +  y\ 


1  -  y  1-2/^ 


a+b      a—h 
a  —  b      a  +  b. 

1  1 


.  (a  +  h 

a-l\ 

'  \a-b 

a  +  b) 

X 

y 

141 


a  —  x      a  —  y      (a  —  x)^      {a  —  y)"^ 

1  1 

{a  —  y)(a  —  x)^      (a  —  x)(^a  —  y)^ 


■X  ,  x-y  ,   C.y-g)(g-a:)(a;-y) 


a^  —  he         .  Ifi—  ca         ,         e^  —  ah 


+ 


+ 


X  y         ,        z 


37.    If  — —  =a, 

3/4-2  a;  +  2 


h. 


x  +  y 


=  (?,  find  the  value  of 


1+a      1+b      1+0 


CHAPTER   IX 

SIMPLE   EQUATIONS 

128.  Some  forms  of  equations  have  already  been  defined 
and  discussed  in  Chapter  IV.  As  before,  §  64,  the  last 
letters  of  the  alphabet  are  used  to  represent  unknown 
quantities,  and  the  first  letters  are  used  to  represent 
known  quantities. 

129  An  integral  equation  is  one  which  does  not  contain 
the  unknown  quantity  in  any  denominator.  A  numerical 
equation  is  one  which  contains  the  unknown  quantities 
and  numerical  quantities  only.  A  literal  equation  is  one 
which  contains  other  literal  quantities  than  the  unknown 
quantity. 

Thus,  2  a;  +  3  =  11  is  both  integral  and  numerical ;  a  +  x  =  b 

2  4 

is  both  integral  and  literal ;  -  +  3  =  -  is  both  fractional  and 

X  X 

numerical ;   -  +  6  =  c  is  both  fractional  and  literal. 

X 

130.  The  degree  of  an  equation  in  one  unknown  quan- 
tity depends  upon  the  highest  degree  which  that  unknown 
quantity  may  have  in  any  term.  If  the  equation,  in  its 
simplest  integral  form,  contains  the  first  degree  of  the 
unknown  number  as  the  highest  degree,  the  equation  is 
said  to  be  of  the  first  degree,  or  a  simple,  or  linear  equation. 

Thus,  ax  +  b  =  c  is  a,  siini)le  equation. 

142 


(H.  IX,  §131]  SIMPLE   EQUATIONS  143 

NUMERICAL   FRACTIONAL   EQUATIONS 

131.  Two  equations  are  said  to  be  equivalent  when  the 
roots  of  the  equations  are  idefitical.  The  general  method 
for  the  solution  of  simple  equations  consists,  as  in  §  68, 
in  the  transformation  of  the  original  equation  into  a 
series  of  equivalent  equations,  until  such  a  simple  form 
s  obtained  that  it  contains  as  a  left  member  only  the  un- 
known quantity,  and  as  a  right  member  only  the  known 
quantity. 

n  .    1       2  a;- 3 


1.  Solve  7^.-ij  +  jL,=  f:^.  (1) 

Simplifying  i,i  (1),       7  a,  -I  + 1  =  ^^.  (2) 

Multiplying  (2)  by  36,  the  L.  C.  M.  of  the  denominators, 

252a;-T(4)+3  =  6(2a;-3),  (3) 

liniplifying  in  (3),       2r)2  x  -  28  +  3  =  12  a;  -  18,  (4) 

ransposing  in  (4),           252  a;  -  12  a.'  -  28  -  3  -  18,  (5) 

uiiting  in  (5),                              240  x  =  7,  (6) 

lividing  (6)  by  240,                           '^  =  2i0*  ^^^ 

2.  Solve    5  +  ^:^-1- =  15.  (1) 

Multiplying  (1)  by  28  x, 

28(8) +4(5  +  0;) -14(3)  =7a;(15),  (2) 
implifying  in  (2), 

224 +  20 +  4,0; -42  =  105  a?,  (3) 
ransposing  in  (3),             4  o;  -  105  x=  -  224  -  20  +  42,         (4) 

niting  in  (4),                          -  101  a;  =  -  202,  (5) 

ividing  (5)  by  -  101,                       a;  =  2.  (6) 


144  ELEMENTARY   ALGEBRA  [Ch.  IX,  §  131 

To  solve  a  simple  equation  in  the  fractional  form  and  con- 
taining one  unknown  quantity  • 

Multiply  every  term  of  each  member  of  the  equation  by  the 
L.  (7.  M  of  the  denominators ;  transpose  the  unknown  terms 
to  the  left  member^  and  the  knoivn  terms  to  the  right  member ; 
unite  similar  terms.  Divide  every  term  of  each  member  of 
the  equation  by  the  coefficient  of  the  unknown  quantity. 

Multiplying  by  the  L.  C.  M.  of  the  denominators  is 
called  clearing  the  equation  of  fractions. 


EXERCISE    LXIV 

Solve  the  following  equations  : 

X        X 

3    42     1       1  _40 

'     X      X     3x      3 

.  u?+?=l. 

XXX 

4.      ^            ^     -1. 

5x     lOz     10 

9  X      X       S  X      9  X 

6.    J-  +  J-  +  -1  +  J— l^=.o. 

Qx     12x     Sx     24:x     72 

,.    4  +  ?-^-UI  =  0. 
X        11  :z:        6      X 


5     97     5(11 -3a;)  ^7-9 


X 


X       1  6x  2x         Ix 

^     Sx  +  1      5:?:-l      ^      9a;  +  5^21      5 
4:x  8x  bx  5       sc 

10.    _L_Jl  +  i3_  =  4_l      3. 
Sx     12a;      l(Ja;      3      6      8 


Ch.  IX,  §  132]  SIMPLE   EQUATIONS  145 

132.    The  method  of  procedure  in  case  the  denominators 
contain  several  terms  is  the  same  as  in  §  131. 

1.    Solve  ^_---ll-  =  13.  (1) 


Factoring  the  denominators  in  (1), 
r  11 


:13,  (2) 


2(4a;  +  l)      6(4a;  +  l) 
multiplying  (2)  by  10  (4  a;  + 1), 

5  (7) -2  (11)  =  13  (10)  (4  a; +  1),  (3) 

simplifying  in  (3),  35  -  22  =  520  x  + 130,  (4) 

transposing  in  (4),  -520  a;  =  -35 +  22 +  130,  (5) 

uniting  in  (5),  -  620  a;  =  117,  (6) 

dividing  (6)  by  -  520,  a;  =  -  ^^^  =  -  ^%.  (7) 

2.    Solve   ^^_^  =  4.  (1) 

x  —  1       x  +  1 

Multiplying  (1)  by  {x  -l){x-\- 1), 

{5x-\-l){x  +  l)-{x-^){x-l)  =  4t{x--l)(x  +  l),     (2) 

simplifying  in  (2), 

5a;2  +  6a;  +  l-a;2^10a;-9  =  4a;2-4,  (3) 

transposing  in  (3), 

5x^-x'-4.x^  +  ^x  +  10x:=-l  +  Q-4.y  (4) 

uniting  in  (4),  16  a?  =  4,  (6) 

dividing  (5)  by  16,  a;  =  J.  (6) 

Although  (3)  contains  x^,  yet  the  equation  can  be  solved  as  a 
simple  equation  because  the  simplified  form,  (5),  contains  only 
the  first  power  of  the  unknown  quantity. 

Note.  Each  term  of  a  fractional  equation  which  is  in  the  frac- 
tional form  should  be  reduced  to  its  lowest  terms. 


146  ELEMENTARY   ALGEBRA  [Oh.  IX,  §  132 


EXERCISE  LXV 


Solve  the  following  equations : 


x  +  2  3(a;  +  2)      3  3(^-7)      6      2a;-14 

2        5              3      ^5  ^^  120      ^x-\0     x  +  \0 

x  +  1  2x  +  2      2'  ■  144-x2      12  +  x      Vl-x 

9                7      _13  ^^  5x    ,  7  +  4a;_-,      6-5a: 


2a;  +  2      3a;  +  3      12  a;-3     4a;-7  a;-3 

1  1^1 

2-3a;     4-6a;      6' 


12. 


5.    JL^— 11-  =  ..  13. 


3a^  +  42 

^  +  5 

1 

6a:2+7a 

^  +  8 

2 

x-^ 
^     10  + 

x-A 

=  2. 

a;  +  l      2a;  +  2      2  x 

4        ,       31      _1         ^^     5a;2+7,y+4_3a.2+63;+7 


2a;  +  2      3a;  +  3      6  Ibx^+x-ij     9a;2+6a;  +  3 

7  13                 7         ^  ^  ^5      7  3^-2      2r?:  +  r>^lly+3 
a;- 2      5a^-10        '  '    5x+3     3a:  +  9     15a;  +  9' 

8  15              ^      ^  11  16     5a:  +  2      3x  +  1^3a:  +  2 
"3  — 2a;     6  — 4x         *  '4a;  +  3      Ga;  +  2      4x-t)' 

1           11  a;                   4 


17. 


18. 


2a;  +  l      12      2a:  +  l      3(2a;  +  l) 

5a:  +  4      3(a;-7)^^3  9 

a;— 1        o(a;— 1)  5(a;— 1) 


19     5_+J  ,2x-l        7a;4-5_a;2-25 


20. 


4  3a;-12      8a;-32      4a:-16 

13a;  +  lQ      2ri0a;+l)       7-lla;^ 
28a;-32       49a;-56        35a;-40 

1  +  a;  ,   r  ,     2x2   _,,      \  +  x 


21.  2::li:i  +  ,5  +  -j^2=3-^,     . 

3^        -^  9       "^  3    '    *^ 


Cii.  IX,§133]  SIMPLE   EQUATIONS  147 

133.  If  the  equation  contains  several  terms  in  one 
(k  nominator  and  several  simple  denominators,  the  pro- 
cess of  solution  is  much  simplified  by  first  multiplying 
every  term  of  each  member  of  the  equation  by  the  L.  CM. 
(){  the  simple  denominators  and  then  simplifying  the 
resulting  equivalent  equation. 

1.    Solve- -^ — = — -^ ^ —  (1) 

x  +  4:  15  5  3  ^  ^ 

Multiplying  every  term  of  each  member  of  (1)  by  15, 

I^-(16x  +  59)  =  3(3x  +  2)-5(5x  +  l),  (2) 

X-\-4: 

simplifying  in  (2), 

X-{-4: 

transposing  integral  terms  in  (3), 

75  a; 


x  +  4: 


=  16a;  +  59  +  9a;  +  6-26a;-5,      (4) 


75  a; 

uniting  integral  terms  in  (4),                     =  60  (5) 

X  ~\~  t: 

dividing  each  member  of  (5)  by  15,         ^  =  4,  (6) 

multiplying  each  member  of  (6)  by  a;  +  4,    5  a;  =  4  a;  + 16;  (7) 

transposing  and  uniting  in  (7),                         a;  =  16.  (8) 


Some  equations  which  appear  to  be  higher  than  first 
degree  equations  may  be  solved,  by  various  devices,  ns 
first  degree  equations. 


148  ELEMENTARY   ALGEBRA  [Ch.  IX,  §  133 

2.  Solve --  = 7 -•  (1) 

x-2     x-8     x-4:     x-~5  ^  ^ 

Uniting  the  members  in  (1)  and  simplifying, 

-1 = ~'~  (2) 

(x-2){x—3)      (x-4:)(x-5y  ^  ^ 

multiplying  each  member  of  (2)  by  the  L.  C.  M., 

(x  _  4)(a;  -  5)  =  (x  -  2)(x  -  3), 

simplifying  in  (3),  0(y^  —  9x-{-20  =  x'^-~5x  +  6 

transposing  and  uniting  in  (4),       —  4  a;  =  —  14, 

dividing  each  member  of  (5)  by  —  4,    x  =  |-. 

3.  Solve  2^1+^  +  ^^^  =  ^^+1^. 

x—1         x—2  :r  +  3 

Eeduce  each  fraction  in  (1)  to  a  mixed  expression, 


x  —  \  X  —  2  a;  +  3 

uniting  integral  terms  in  (2), 

5      .     10    ^    15 
a;-l"^a;-2     .^  +  3' 

dividing  every  term  of  each  member  of  (3)  by  5, 

12  3 


a;-l      a?-2     aj  +  3'  ^  ^ 

multiplying  every  term  of  each  member  of  (4)  by  the  L.  C.  M., 

(^_2)(a^  +  3)  +  2(a:-l)(a;  +  3)=:3(a;-l)(a;-2),  (5) 
simplifying  in  (5), 

a;2_|-a;_6  +  2a^  +  4T-6  =  3a;2_9aj  +  &  ('6) 

transposing  and  uniting  in  (6),          14x  =  18,  (7) 

dividing  each  men^ber  of  (7)  by  14,      a;  =  -f-.  (8) 


:ii.  IX,  §  133]  SIMPLE   EQUATIONS  149 

EXERCISE  LXVI 

Solve  the  following  equations : 

^     2x+7      Sx  +  8__4:x+J^ 


2. 


3. 


5. 


6. 


10. 


11. 


12. 


4 

5x  +  2           8 

Qx  +  l 

lla;-l_2x  +  ll 

15 

4a;+3             5 

5x  +  l 

3x  +  2      15a;-39 

12 

5a;-8            36 

2a;+3 

2a;-l      x  +  S 

10         42^+2         5 
a:j— 3      x  —  4:     x—5     x  —  6 


x  —  4:     x—5     X—  Q     x—1 

X+1  _  X+2  ^  X+8  __  X  +  4: 

x  +  2     x  +  S'^  x  +  4:     x  +  5 

21a;  +  13      8a;  +  13^g 
3^+1         4:i;  +  l 

9x+2      5x+2^Q 
Sx  —  1       x  +  1 

2ir  +  3  ,  4:x  +  5 


=  6. 


x  —  4:  X  —  Q 

x  +  8  __x  +  4  ^  x  +  5  __  x+  Q 
x+4     x+5     x+ 6      x+ 7 

4a;  +  5  _  14a;+3  ^  16a:  +  3 
9  35a^+l  36 

X 


-10      13a;-2      47  2:-!      6rr  +  7^. 

7  10^  +  7  35  5 


13      3rr+8      5a:  +  8      10a;  +  27  ^  ^g 
'a;  +  l         :i:  +  2  a;  +  3 


150                                ELEMENTARY   ALGEBRA             [Ch.  IX,  §  134 

LITERAL   FRACTIONAL  EQUATIONS 

134.    Literal    fractional   equations   are    solved    by  the 
Rule  given   in   §  131. 

1.  Solve  a  +  ^-  =  ^-h.  m 

XX 

Multiplying  each  term  of  (1)  hj  x,      ax-{-b  =  a  —  hx,  (2) 

transposing  in  (2),                                     ax  +  hx  =  a—  h,  (3) 

factoring  in  (3),                                        x{a-\-'b)  =  a—'bf  (4) 

dividing  each  member  of  (4)  by  a  +  6,             x  =  ^^^ —  (5) 

2.  Solve  ^^^^^  +  d  =  x  +  a.  (U 

c  ^  ^ 

Multiplying  (1)  by  c,     ax—b  +  cd  =  cx  +  ac,  (2) 

transposing  in  (2),                     ax—  cx  =  b  —  cd  +  ac,  (3) 

factoring  in  (3),                       x(a  —  c)  =  b  —  cd  +  ac,  (4) 

b    —Cd   +   aC  yr^. 

x  = •  (5) 

a  —  c  ^ 

1.  (1) 

Multiplying  (1)  by  a^—b'^, 

bx=  (x+3  b)(a-b)  -  (a^-b'),  (2) 

simplifying  in  (2),                 bx=ax—bx^3ab—3b^—a^-{-b'^,  (3) 

transposing  iu  (3),  bx—ax-\-bx=3  ab—3  b^—a^-{-b^,  (4) 

uniting  in  (4),            2bx  —  ax  =  —  a^  +  3ab  —  2b^y  (5) 

factoring  in  (5),        x(2  b  —  a)=  —  (a  —  b)(a  —  2b)j  (6) 

dividing  (6)  by  2  6  -  a,          x  =  -  (a  -  5)(-- 1),  (7) 

simplifying  in  (7),                  x  =  a  —  b.  (8) 


dividing  (4)  hj  a  —  c, 

3.    Solve     Z"^,  = 

x  +  Sb 
a  +  b 

Ch.  IX,§134]  SIMPLE   EQUATIONS  151 

EXERCISE  LXVII 

Solve  the  following  equations  : 

ax     M  ^      m  ,   n 

1.  —=1.  5.     —+—  =  (?. 

ax     ox 

^^  i^  V  ^^         X     ,    X    ,    X 

2.  —  =  -•  6.    —  H f-  =  a. 

m     n     p 

^     w      -,      ^      ^  ax      bx      ex       . 

3.  --1  = 7.    — +  — +  — =  c?. 

"  -  m       n      p 

a         b        e        ^ 
8.    —  +  — 4._  =  ^. 
ma:      nx     xp 


ax 

T'' 

=  1. 

a 
bx'' 

_b^ 

a 

a 

1  = 

a  — 

1 

x 

X 

X   ,^_ 

-  a. 

m 

n 

I 


9.     ^4.A  +  _^=a2  +  52  +  ^. 

6(?a;      acx      aox 

,^     ab  ,  ae  ,   be       1,1,1 
10.    _++_=      +      +. 

ex      ox      ax      a-      o^      c^ 

^^     rr+^  ,   7  __  2(x  -  b)  ^  ^ 
b         3  Sa 

^2     SCx-4b)      5Cx-Sa)^^ 
a  46 


__     ax  —  l  .  bx  —  la^  + 

13. [■ 


14. 


15. 


b  a           €^b^ 

ex  +  ab  j_ax+^     cfi^ 

a  be 

b      b'^  a        1       a^ 


ax 


d^     2bx     2     2  62 


b-\-e     a  +  e     a  +  b_2 
box       aex       abx      c 

__       ax        1 

17.    r  =  l— ;c. 


18 

X         b  +  x 

a  +  b         a 

19. 

a  +  x      ^__  1 
b  +  e      a 

20. 

ab  —  c^x       a  __  ex 
a-b        Ze~3b 

21. 

X                a 

b  +  ex      e(l  +  a} 

22. 

ax           bx          9 
=  a^  - 

a—b  a—b      a+b 


152  ELEMENTARY   ALGEBRA  [Ch.  IX,  §  134 

X  —  2  a  +  Sb  _      5  b  ax  +  be  _bx  —  ac  _^c 

'    x  +  2a  —  Sb'~  ia  —  b  '      b  +  c        d^-^V^~  a 

X  —  a  .  x-\-3b      Q  „^     ctx^  +  bx  +  1      a 

a  —  5       a  +  b  bx^  +  ax  +  1      b 

X  9      X—  2bc 


28. 


31. 


33. 


34. 


36. 


fx 

a[ i 


27.    a  --b 


a  -{-b  a  —  b 


2bx      ,  ^^  —  ^  __  ^(^  —  2  ab^  ___         7 
ab  +  cd  a  ab  —  cd 


^     X  —  ab  ,  X—  ac  ,  x  —  bc      o      be  +  a^  ,  c 

29. 1 H d  = ■  +  -. 

ae  be  ab  ac         0 

(b  —  e)x  ,     ex  +  b^       2b  —  c 

30.  v^ ^  -I = . 


bx  —  e^       e(x  +  J)  b 

ax  +  be      bx  +  a{e  —  1)  _  t^  —  5  ,  (a  —  5)(a;  —  g)  +  5 
bx  +  ae  ax—  be  a  bx  +  ae 


a         ,  X  —  ae      1      bx—  a<?     ac 
32.    -; -  + 


2x  —  ab       b  —  e       e      b(b  —  (?)       b 

Sa  —  x      2{x—  g)      x-\-2b  __  -1  __  £ 
~~b  3a-26      3a  +  g~         h 

ax  +  b        ax—e   __b^ -\- c(a +  b') 
bx  +  c      bx  +  2c~'    b(^bx  +  8e} 


^^     abx  +  1   ,  acx  —  1  ,  bex  —  1      ab  +  ae  +  be 

35.      —   -\ 1 :- = • 

a  +  0  a  —  c         b  —  e  abe 

ax  +  b  _  bQax-by  ^  2b(b  -  a^(2b  +  a) 
x  +  l         bx-a    ~  2b(bx-a^  +  b^-a^' 

^^     a(x-2e^      b{x-2a^  ^  <^  +  25)^^^ 
a— b  —  e        e  —  a  —  h        a  +  b  +  e 

2a  —  x,2b  —  x  2  e  —  x  x 

38.       ; h 


a—b—e      b—a—e      c—a—b      a+b+c 


Ch.  IX,§135]  SIMPLE   EQUATIONS  l58 

135.  A  quantity  q  is  said  to  be  expressed  in  terms  of  the 
quantities  m  and  n  when  q  is  the  left  member  of  an  equation 
which  contains  m  and  /^  and  numerical  quantities  only  in 
the  right  member. 

Thus,  \i  a—h  +  Cy  a  is  expressed  in  terms  of  h  and  c. 

EXERCISE   LXVIII 

In  the  following  equations  express  each  literal  quantity 
in  terms  of  the  other  literal  quantities  : 


1. 

a      c 
h~  d 

2. 

X  __  h 

r     a 

3. 

T 

1       1 

4. 

a'    ah' 

5. 

1         2 
F     a  +  h' 

6. 

^  +  1  =  1. 

7. 

„  _  (f,  -  v„)« 

'-        2       • 

Q 

V.P^^V.P. 

o* 

T,           T,   ' 

9. 

tv  —  tw  =  t  —  1. 

10, 

(7=|(i^-32). 

11. 

Ma  —  Jib  =  a  —  b. 

12. 

13. 

aoc 

14. 

ax  =  bw. 

15. 

V=  abc. 

16. 

V,  p; 

17. 

L 

18. 

abc  =  4  R$. 

19. 

^1  —  V(. 

20.  T='^(h  +  h'y 

21.    c^  =  |(2«  +  l). 

22     mp  —  rs s  —  rp 

w  n 


154  ELEMENTARY  ALGEBRA    [Ch.  IX,  §§  136, 137 

INDETERMINATE   EQUATIONS 

136.  If  a  single  simple  equation  contains  two  unknown 
quantities,  there  is  an  infinite  number  of  solutions  :  hence, 
fiuch  an  equation  is  called  indeterminate. 

Thus,  x-\-y  —  iy  is  a  simple  indeterminate  equation.  If  in 
x-\-y  -=0,  x  =  0,  then  2/  =  5 ;  if  x  =  \,  then  ?/  =  4|- ;  if  a;  =  2, 
then  ^  =  3 ;  and  so  on. 

137.  The  solutions  of  indeterminate  equations  are  often 
restricted  to  those  in  which  the  roots  are  both  positive  and 
integral.     Such  solutions  can  often  be  found  by  inspection. 

1.  Find  the  positive  integral  solutions  of  2  a;  +  5  ^Z  =  14. 

If  2/  =  0,  0^  =  7;  if  y=l,  x  =  4.\',  if  y=2,  a;=2;  if  2/=3, 
x=     Y* 

Whence  the  positive  integral  solutions  are :  y  =  Oy  x=  7 ; 
y  =  2,x  =  2. 

2.  Find  the  positive  integral  solutions  of  2x  +  31/  =  19. 
liy=0^x  =  9^',  if  ?/=l,  .T=8;  if  ?/=2,  ct'=6i;  if  2/=:3,  a;=5; 
ii  y=4:,x=S^;iiy=5,x=2i  iiy=6,x=^i  ify=7,x=-t 
Whence  the  positive  integral  solutions  are ; 

2/  =  l,  a;  =  8;  7/  =  3,  0^  =  5;  y  =  5,x=z2, 

EXERCISE   LXIX 

Find  the  positive  integral  solutions  of  the  following  : 

1.  7  a: +  5  ^=38.  4.   2:^  +  17^=70.  i 

2.  6:r  +  ll^  =  125.  5.    32  2:  + 3^  =  1624. 

3.  a;  +  20«/=r53,  6.   11  a;  =  576  -  13  «/. 


Ch.  IX,  §  137]  SIMPLE   EQUATIONS  155 

7.  Ux-9y  =  l.  ^3    §^  +  5^=36. 

8.  8:?; -15^  =  33.  ^         ^ 

9.  a;- 10^  =  6  14.   — - — +    ^^T    =7. 

o  lU 

10.  11  2:=  7  ^+114.  ^  ^       ^ 

15    3^  +  5^7a;  +  y-6 

11.  9  a:  =11^.  *         4  8 

-.o     3:r      rr  on  ,^     3(:r  +  4  y  -  50)      ^ 

12.  -—  =  7  V  —  29.  16.    ,\     — ^ — ^  =  1. 

4  "^  19a:  +  7/-200 


Find  the  least  positive  solutions  of  the  following : 
17.     -—-  +  —f^  =  11.  20. '-  =  —  ^^ 

4         3  9         4  2 

18  3rr+7^4rr-y  +  l  ^i     ^^Zl^jL^^l 

4  2  '    2^/-a;+5 

19  4a;      3y^2:r+y+2  3(2:r-y  +  2)^. 
'3^8                2                   •    4a;-32/+10        * 

23.  In  how  many  ways  can  $110  be  made  up  of  ten- 
dollar  bills  and  two-dohar  bills  ? 

24.  If  A  spends  76  cents  in  buying  pencils  at  3  cents 
each,  and  penholders  at  2  cents  each,  how  many  of  each 
does  he  buy  ? 

25.  How  many  golf  balls  at  50  cents  each,  and  how 
many  baseballs  at  $1.25  each,  can  be  bought  for  $9? 

26.  How  many  baseballs  at  $1.25  each,  and  how  many 
baseball  bats  at  75  cents  each  can  be  bought  for  $  21  ? 

27.  In  how  many  ways  can  railroad  stocks  at  $105  and 
$95  respectively  per  share  be  bought  for  $5900  ? 


156  ELEMENTARY   ALGEBRA  [Ch.  IX,  §  137 

REVIEW  EXERCISE   LXX 

Solve  the  following  equations : 

1.    T:^^ll^fll  =  8x  +  7. 
4 

2. 2£±i_r._^V36 


4. 


2  V  7 

x  +  S      a:-2^3x-5      1 
2  3  12         4" 

5x-l_8x-2^5-x 
8  7  4    ■ 


„     „        x  —  4:      ,      5x  +  14 
5.    3:.--^-4  =  -^-. 

2rr  +  l      42;H-5^2a:+5      ri:+8 
3  4  8  6     ^ 

X—  a       S  X—  c 


:=0. 


2  i^;—  6      6  X—  d 


g     £-2^_13a2_^^ 

a;  H-  3  a        :?;2  —  9  a^ 

a:  +  (a  —  b')x  __cx—  d 
a—  b  c 

ah  +  x      IP'—  X  __x—h      ah-^x 


11.    ^^  +  4J  =  -     ""^ 


a  —  b  3  a  +  S 

.0^.5  4 

12. h 


l-3x     1_5^     l-2a; 

13     5_+2_a^  (g;  +  aY 
x-'lb      (x-bY 


Ch.ix,§i373  simple  equations  157 

'    1x-\      3:^  +  2      6* 

hx+\      a(x^-Y) 
15.    ax ' —  =  -^^ . 

X  X 


16. 


17. 


19. 


21. 


22. 


23. 


24. 


25. 


a;^  +  a^  a:  1 


^x^—a^      2x  +  a  4 

1  2 


2{Sx+l)      32;2  +  22a;+35      2ir  +  10 


18.    #+4-4      ^" 


3a;  +  l     5-6:?;     5  +  9:z;-18:c2 

a     ^     h     __}?"—  a? 
X—  a     x—h      W'—hx 


a  —  X  ,  h  —  x\  c—  X      r. 

20.    — 1 H =  0. 

OG  ca  ab 


ax—h      hx—a  a  —  h 


ax  +  h      hx  +  a      (ax  +  h)(hx  +  a) 

a^  +  4:  a  ^     _         1 

x^  +  x—a^+  a      x  +  a     x—a  +  1 

6x-{-l       2x-4:  _2x-l 
15  Ix-1Q~       5      * 

8^+1      1 X-&  ^^x+2      1 
18         bx-4t  9  6' 

h(x  —  li)  a(x  —  ^)  g(a;  —  g)     __  a; 

2a  +  5-!-2(?     a  +  2b  +  2c      2a  +  2h  +  o'^2 


CHAPTER  X 


GRAPHS 


138.  In  Chapter  I  it  was  shown  that  numbers  can  be 
represented  by  distances  along  a  line  from  a  given  fixed 
starting-point.  In  the  present  chapter  it  will  be  shown 
how  the  relation  between  two  algebraic  quantities,  which 
are  connected  in  any  way,  can  be  represented  by  draw- 
ings to  scale.  As  a  first  step  in  this  direction,  it  is  neces- 
sary to  establish  certain  conventions,  by  the  aid  of  which 
the  position  of  any  point  in  a  single  flat  surface,  or  plane, 
can  be  fixed  by  two  algebraic  quantities. 

139.  Constructing  a  pair  of  perpendicular  lines,  called 
axes,  X^ X  and  Y'  P",  as  shown  in  Fig.  3,  a  point  can  be 
located  by  saying  that  it  is  m  units  above  or  below  X' X^ 
and  n  units  to  the  right  or  left  of  Y'  Y. 

j^Y  If,    instead   of   using 

the  words  "above"  and 
"  below,"  "  right  "  or 
"  left,"  it  is  understood 
that  all  distances  meas- 
ured upward  or  to  the 
right  are  positive^  and 
those  measured  down- 
ward or  to  the  left  are 
negative^  tivo  numbers 
with  the  proper  signs 
attached    will    represent 


N 


M 


Fig.  3. 


158 


Ch.  X,  §§  140-141] 


GRAPHS 


159 


the  distances  of  the  point  from  the  two  lines^  and  these  two 
numbers  taken  together  will  locate  absolutely  the  position  of 
any  point  m  the  same  plane  with  the  lines, 

140.  The  distance  of  a  point  to  the  right  or  left  of  Y'  Y 
is  given  first  ;  and  the  distance  of  this  point  above  or 
below  X' X  is  given  second.  These  two  distances  are 
called  the  co'drdinates  of  the  point.  The  coordinates  are 
written  in  parenthesis  ;  thus,  P  =  (3,  4)  means  that  the 
point  P  is  3  units  to  the  right  of  the  vertical  line  Y'  Z", 
and  4  units  above  the  horizontal  line  X' X. 

141.  Any  point  whose  coordinates  are  known  can  be 
definitely  located. 

Thus  the  locations  of 
the  points,  A  =  (3,  4), 
B^{^2,  6),  (7=  (-5, 
-7),  D  =  (6,  -  3),  are 
shown  in  Fig.  4. 

If  either  of  the  coor- 
dinates is  0,  the  point 
will  lie  on  one  of  the 
axes. 

Thus,  the  location  of 
E  =  (0,  -  5)  and  of 
F  =  (2,  0)  is  shown. 


Y 

D 

X 

0 

F 

X 

D 

E 

c 

^ 

y 

Fig.  4. 


EXERCISE    LXXI 

Locate  the  following  points  whose  coordinates  are  : 

1.  (2,4).  4.    (-4,-8).  7.    (-4,-31). 

2.  (-3,4).  5.    (0,-9).  8.    (-3,21). 

3.  (-3,-4).  6.    (0,0).  9.    (-7-^,9). 


160  ELEMENTARY   ALGEBRA      [Ch.  X,  §§  142,  143 

GRAPHS   OF   SIMPLE   EQUATIONS   IN   ONE   UNKNOWN 
QUANTITY 

142o  If  a  single  equation  in  x  and  y  is  given,  it  is 
evident  that  the  coordinates  of  points  taken  at  random 
will  not  satisfy  it,  since,  if  a  value  is  assigned  to  one  of 
the  coordinates,  the  other  will  be  determined  by  such  an 
equation.  There  are  then  only  certain  points  whose 
coordinates  satisfy  the  given  equation,  and  it  will  be 
discovered  that  these  points  lie  consecutively,  and  hence 
form  a  curve  (or  straight  line).  Such  a  curve,  which 
contains  all  the  points  which  satisfy  a  given  equation,  is 
called  the  graph  of  that  equation. 

143.  In  case  the  equation  contains  only  one  unknown 
quantity,  as  x=  5^  the  graph  is  very  easily  determined, 
since  the  equation  says  that  every  point  which  satisfies 
it  must  have  its  :z;-coordinate  equal  to  5,  but  places  nf^ 
restriction  upon  the  ^/-coordinate.  All  such  points  lie 
in  MJV,  Fig.  5,  5  units  to  the  right  of  the  axis  Y^  Z",  and 
MJV  is,  therefore,  the  graph  of  the  equation,  x=  5.  Simi- 
larly, the  graph  of  any  simple  equation  in  one  unknowu 
can  be  shown  to  be  a  line  parallel  to  one  of  the  axes. 


EXERCISE 

;  Lxxii 

Construct  the  graphs  of  the 

equations  : 

1.  x-\-5  =  6. 

2.  ^  +  4  =  9. 

e.  1  +  1  =  2. 

3.    9-{-5x==lQ  +  4:x. 

'•  M=»- 

4.  82/  =  5  +  102/-ll. 

5.  5:^;- (3a;- 7)  =17. 

„     Sx+5     X 

13- 4x 

2 

Ch.  X,  §  144] 


GRAPHS 


161 


GRAPHS    OF   SIMPLE   EQUATIONS   IN   TWO   UNKNOWNS 

144.  As  tlie  simplest  type  of  equations  in  two  un- 
knowns, consider  those  in  which  the  known  quantity  is 
wanting.  Any  such  equation  may  be  put  into  the  form, 
y  ==  ax^  where  a  can  have  any  value  —  positive,  negative, 
I  or  fractionaL  All  points  whose  coordinates  satisfy  this 
I  equation  must  have  their  ?/-co5rdinate  a  times  their 
x-coordinate,  and  hence  must  lie  on  a  straight  line 
through  the  origin,  as  KL  in  Fig.  5.  To  determine 
the  graph  of  any  such  equation,  plot  any  one  point  which 
satisfies  it,  and  draw  a 
line  of  indefinite  length 
through  this  point  and 
'the  origin. 

For  example,  the  equa- 
;  tion  Zx  =  2  y  is  satisfied 
[by  (2,  3).  Hence  a  line 
[  through  P  =  (2,  3)  and 
j  (0, 0)  is  its  graph.     If  the 

pupil  has  not  had  enough 

geometry  to  be  sure  that 

all  points  whose  coordi- 
nates satisfy  the  equation 

must  lie  on  the  line,  let 

him  plot  a  number  of  such  points,  as  (1,  |-),  (3,  |),  (4,  6)p  etc.. 

and  convince  himself  that  they  all  do  lie  on  the  line. 


Y 

A 

/i 

/ 

W 

?y 

v//J|\ 

f  ■ 

"          / 

* 

r: 

Yy' 

x' 

( 

'/v 

X 

>      4 

/ ' 

1    -•  r 
!  / 

/ 

'      /^^ 

J 

7 

i 

7 

/ 

/ 

kj 

/ 

r 

M 

EXERCISE    LXXIII 

Construct  the  graphs  of  the  following  equations  : 

1.     y=^X,  3.     X+9/=:0,  5,      SX+8  9J=0. 

2.   5y  =  lx.  4.   2a;— 5y=0.       e.  x—5y  =  0. 


162 


ELEMENTARY  ALGEBRA 


[Ch.  X,  §  14( 


145.  Any  simple  equation  in  x  and  y^  wliicli  contains: 
a  known  quantity,  can  be  reduced  to  the  form  y  —  ax  +  h, 
If  the  graph  of  the  equation  y  —  ax  is  plotted,  and  f romi 
every  point  on  this  line  lines  parallel  to  ^  Z'and  equal  in 
length  to  h  are  drawn,  the  extremities  of  these  lines  will 
evidently  be  the  points  whose  coordinates  satisfy  the  equa- 
tion y  =  ax-\-h.  These  points  are  also  on  a  straight  line. 
It  will  be  noticed  that  the  graph  of  every  equation  of  the 
first  degree  in  x  and  y  is  a  straight  line.  To  find  the 
graph,  it  is  only  necessary  to  determine  two  points  and 
draw  a  line  through  them.     These  two  points  are  usually 

taken  on  the  axes. 

For  example,  the  equa- 
tion 2a;— 32/  +  6  =  0  is 
satisfied  by  (—3,  0)  and 
(0,  2) ;  its  graph  has  the 
position  of  MN  in  Fig.  6. 
The  pupil  should  assure 
himself  by  trial  that  this 
line  contains  every  point 
which  satisfies  the  given 
equation ;  for  example, 
the  points  (1,  2|),  (2,  3i), 


J 

K 

X 

\ 

f/ 

X 

h 

y 

^ 

\y^  1 

y 

.^;,<>^    !    > 

y 

^!^ 

< 

y 

■^ 

"r 

A 

/" 

>    0 

X 

X 

\y 

^  \  y 

y 

1 

^ 

^ 

y 

y 

^ 

^ 

y 

^M 

y 

y 

K 

Y\ 

Fig.  G. 


(4,  4|),  (-1,11),  etc. 


EXERCISE   L.XXIV 

Construct  the  graphs  of  the  following  equations  z 

1.  x  +  y  =  8.  5.    8x-\-4:y  =  21. 

2.  x  +  Dy=  16.  6.    4:x-^5y=  25. 


16. 

3.  4  2:  +  ^  =  10. 

4.  Sx+2y=:13. 


7.  x-\-()y  =  20. 

8.  Sx  +  2y  =  24:. 


CHAPTER   XI 

SIMULTANEOUS   SIMPLE  EQUATIONS 

146.  Two  or  more  simultaneous  equations  are  those 
Whioh  can  be  satisfied  by  the  same  values  of  the 
unknowns. 

Thus,  2  X  +  3  y  =  S,  and  3x  +  2  y  =  7,  are  simultaneous  sim- 
ple equations,  since  each  equation  is  satisfied  if  x=l,  and2/=2. 
Similarly,  x  +  y-\-  z=6,  2x  —  y  +  z=3,  and  3x  +  2y—4:  z  =  —5, 
are  simultaneous  simple  equations,  since  each  equation  is  satis- 
fied ii  x  =  l,  y  =  2,  and  z  =  3. 

147.  Two  or  more  equations  are  inconsistent  when  they 
cannot  be  satisfied  by  the  same  values  of  the  unknowns. 

Thus,  x+y=5,  and  x+y=4:f  are  inconsistent  equations,  since 
the  unknowns  cannot  have  the  same  values  in  both  equations. 

148.  Two  or  more  equations  are  dependent  when  each 
equation  can  be  derived  from  the  others. 

Thus,  x  +  y==4:,  and  2  a;  +  2  2/  =  8,  are  dependent  equations, 
since  when  the  second  equation  is  divided  by  2  it  gives  a;+?/=4, 
identical  with  the  first  equation. 

Dependent  equations,  though  simultaneous,  are  redu- 
cible to  a  single  indeterminate  equation. 

149.  Two  or  more  equations  are  independent  when  none 
of  them  can  be  derived  from  the  others. 

Thus,  2  X  +  y  =  5)  and  x  +  3  y  =  10,  are  independent  since 
neither  can  be  derived  from  the  other. 

163 


164 


ELEMENTARY   ALGEBRA       [Ch.  XI,  §§  150,,  151 


150.  A  system  of  equations  is  a  group  of  two  or  more 
equations. 

A  solution  of  a  system  of  equations  is  a  set  of  numbers 
which  satisfy  each  of  the  equations  in  that  system.  The 
process  of  finding  the  solution  of  a  system  of  equations 
is  called  solving  the  equations. 


GRAPHS   OF   SIMULTANEOUS   SIMPLE   EQUATIONS 

151.  Graphs  of  simultaneous  simple  equations  in  two 
unknowns  can  be  constructed  by  the  method  of  §§  144  and 
145. 

Consider  the  simultaneous  simple  equations : 

r^+2y  =  4,  (1) 

U+    2/ =  5.  (2) 

In  (1),  B  =  (0,  2),  A  =  (4,  0)  ;  in  (2),  D  =  (0,  5),  C=  (5,  0). 

In  Fig.  7,  the  location  of 
the  points  A  and  B  gives 
the  line  AB]  and  the  loca- 
tion of  the  points  (7  and  i> 
gives  the  line  CD.  The 
lines  AB  and  OD  cross,  or 
intersect,  at  F ;  and  since 
P  is  on  both  lines,  its 
coordinates  must  satisfy 
both  equations.  Hence 
its  coordinates  are  the 
values  of  x  and  y  which 
would  be  determined  by 
solving  the  two  equations 
simultaneously.  These 
are  found  by  measurement  to  be  x  =  6  and  y  =  —  l.  Two 
lines  which  intersect  represent  simultaneous  equations  which 
have  a  single  solution. 


's 

-n 

- 

Y 

N 

\ 

•v 

X 

\ 

^ 

^ 

\ 

J) 

^ 

*s 

\ 

•?i 

V 

V, 

\ 

V. 

^ 

^ 

b\ 

\ 

*>a 

h^ 

^J 

\ 

JC 

•^ 

SI 

k 

c 

X 

0 

^ 

<: 

^ 

P 

s 

\ 

"v 

V, 

\ 

\ 

Y 

Fig.  7. 


Ch.  XI,  §§  152, 153]     SIMULTANEOUS  SIMPLE  EQUATIONS     165 


GRAPHS   OF   TWO   INCONSISTENT   SIMPLE   EQUATIONS 

152.    Inconsistent  equations,  §  147,  may  be  shown  to 
have  no  common  solution  by  constructing  their  graphs. 

Thus,  find  a  solution, 
if  possible,  of 

'2x  +  y  =  4.,       (1) 

\2x  +  y  =  S.       (2) 

In  (1),  a  B  =  (0,  4), 
^=(2,0);in(2),D=(0,8), 
(7=(4,0). 

In  Fig.  8  the  graphs  of 
(1)  and  (2)  are  such  that 
they  never  meet ;  that  is, 
AB  and  CD  are  parallel 
lines.  Hence  there  is  evi- 
dently no  common  solu- 
tion of  (1)  and  (2). 


V    ^  ^ - 

\  ^^> 

^    s^ 

5   5 

^    ^ 

^^-^ 

v^V 

5^fe 

^1  V^ 

J                                    0  \jyJ  \C                 X 

"  i            v=^ 

3  3 

V   V 

3   3 

r  r 

3   3 

^   ^ 

\    \ 

i            ^   ^ 

Y                        \       \ 

Fig.  8. 


GRAPHS   OF   TWO   DEPENDENT  EQUATIONS 

153.  Two  dependent  equations,  §  148,  may  be  shown 
to  be  reducible  to  a  single  indeterminate  simple  equation 
by  constructing  their  graphs. 

Thus,  find  a  solution,  if  possible,  of 

2x  +  3y  =  S,  (1) 

(2) 


3     2     3 


In  (1),  B  =  (0,  I),  A  =  (4,  0)  ;  in  (2),  D  =  (0,  |),  C=  (4,  0). 

Since  B  and  D  and  A  and  C  have  respectively  the  same 
coordinates,  the  graph  is  a  single  line ;  and  the  given  equations 
are  therefore  reducible  to  an  indeterminate  simple  equation 
whose  graph  has  been  shown,  §  145,  to  be  a  line.crossing  the  axes. 


166 


ELEMENTAKY  ALGEBRA 


[Cii.  XI,  §  153 


EXERCISE 

Determine  the  nature  of  the 
tions  by  the  graphical  method : 


LXXV 

following  systems  of  equa- 


1. 


2. 


3. 


5. 


6. 


8. 


9. 


10. 


11. 


12. 


r  ^  +  ^/  =  8, 

-  2:  4-  3/  =  1, 

x  +  y  =  14, 

2x  +  y  =6. 
2x+Sy  =  12, 
^x+5y  =  20. 
x  +  y=^5, 
2x  +  y=6. 
8x  +  2y=7, 

[2x+Sy  =  8. 
2x-Sy  =  2, 
X  —  5  y  =  —  5. 
2x  +  3y  =  5, 
Sx  +  2y  =  5. 
x-2y  =  4:, 

[2x-'4:y  =  8. 

^5x-Sy=--2, 
4:x+2y=-6. 


13. 


14. 


15. 


16. 


17. 


18. 


19. 


20. 


21. 


22. 


23. 


24. 


Sx+2y=6, 
9a;+6^  =  18. 
Sx-\-4:y=% 
8x+4y  =  12. 

{4:x+2y  =  8, 

2x-Sy  =  2i, 
2x-8y=::6. 
2x+5y=:10, 
ix-Sy  =  12. 
(  5  X  —  6  y  =  S^ 
[l0x-12y  =  6. 
2x-8y  =  0, 
8  a;  -  4  ^  =  0. 

4:x  —  5  y  =  1^ 

^5  x  —  4:  y=9. 

5x  +  S  y  =  5, 

9x  +  4:y  =  9. 

r6x-5y=-l, 

.4  a:+  3^  =  6. 
Sx-5y=12, 
x-10y  =  24:. 


It; 


Ch.  XT,  §§  164, 155]     SIMULTANEOUS  SIMPLE   EQUATIONS      167 

154.  Elimination  of  one  of  two  or  more  unknowns  in  a 
l^ystem  of  simultaneous  equations  is  the  process  of  com- 
ining  the  equations  in  such  a  way  as  to  obtain  fewer 
quations  containing  less  unknown  quantities.  The  quan- 
ity  which  has  been  caused  to  disappear  is  said  to  be 
liminated. 

TWO   UNKNOWN   QUANTITIES 
I.   Elimination  by  Addition  or  Subtraction 


1   155.   1.   Solve  1     ^-  +  2,  =  12, 

(1) 

(2) 

Multiplying  (1)  by  2,                      6  a;  +  4  ?/  =  24, 

(3) 

multiplying  (2)  by  3,                      _  6  a;  +  9  ?/  =  15, 

(4) 

,dding  (3)  and  (4),                                     13  ?/  =  39, 

(5) 

Lividing  (5)  by  13,                                         2/=   3. 

■     (6) 

Substituting  y  from  (6)  in  (1),         3  a;  +  6  =  12, 

(7) 

ransposing  in  (7),                                       3  a;  =    6, 

(8) 

ividing(8)  by  3,                                          x=  2. 

(9) 

Verification:             6  +  6  =  12;  —4  +  9=   5. 

The  above  equations  can  be  solved  by  this  method  by 
ultiplying  the  first  equation  by  3  and  the  second  equa- 

ion  by  2,  and  subtracting  the  equivalent  equations  thus 

.erived. 
It  is  to  be  noticed  that  the  equations  are  checked  by 

ubstituting  the  values  of  the  unknowns  in  the  original 

quations. 


168  ELEMENTARY   ALGEBRA  [Ch.  XI,  §  Kd 

2.   Solve!  11  ^  +  ^^^  =  2^'  (1) 

l9^  +  y=8.  (2) 

Multiplying  (2)  by  2,  l^x  +  2y  =  16,  (3) 

rewriting  (1),  llx-\-2y  =  23,  (4) 

subtracting  (4)  from  (3),  1  x  =  -l,  (5) 

dividing  (5)  by  7,  x  =  -l.  (6ji 

Substituting  x  from  (6)  in  (1),  -  11  +  2  2/  =  23,  (7) 

transposing  in  (7),  2  2/  =  34,  (8) 

dividing  (8)  by  2,  ^  =  17.  (9>| 

Verification:    -11 +34  =  23;  -9  +  17=   8. 

The  above  equations  can  be  solved  by  this  method  by 
multiplying  the  first  equation  by  9  and  the  second  equa^i 
tion  by  11,  and  subtracting  the  equivalent  equations  thusi 
derived. 

That  unknown  is  preferably  chosen  for  eliminatior 
whose  coefficients  are  such  that  they  can  be  made  equa! 
by  the  smaller  multipliers. 

Rule  for  Elimination  by  Addition  or  Subtraction :    Mah 

equal  the  coefficients  of  one  of  the  unknowns  in  each  equatio7 
hy  multiplying  one  or  both  of  the  equations  by  the  necessan 
numbers.  Add  or  subtract  the  resulting  equations  accordim 
as  the  equal  coefficients  have  unlike  or  like  signs.  Find  th 
other  unknown  number  by  substituting  the  value  of  the  un 
known  already  found  in  that  one  of  the  given  equations  whic) 
has  the  least  coefficients,  Ve7nfy  the  solution  by  substitution 
in  each  of  the  given  equations. 


Cn.XI,§155]     SIMULTANEOUS   SIMPLE   EQUATIONS 


169 


6. 


7. 


10. 


11 


12. 


3. 


4. 


EXERCISE 

Solve  the  following  systems 

^     (2x  +  y  =  l, 

I -2a; +  3 3/ =  13. 

7  X  -  3 1/  =  15, 

5a;  +  6y  =  27. 

8  a;  +  17  «/  =  42, 
2a:+193/  =  40. 

4  a; +  63/ =  40, 
6  a;  —  7  «/  =  2. 

17a;-18i/  =  15, 
5a;  +  12y  =  39. 

28  a;  +  y  =  33, 

-  21  a;  +  11  ^  =  34. 

|33:r-(y+9)  =  23, 
l44a;+3(y  +  l)  =  50. 

3a;-7«/  =  l, 
5a;  +  3«/=2. 

9a;-6«/  =  2, 
45a;  +  8  =  72^. 

f  19a; -16?/ =  91, 
■    I  27  a; -20  3/ =130. 

|8a;-9?/  =  34, 

i9a:-8^  =  17. 


LXXVI 

of  equations : 

f6a;  +  5«  =  68, 
13.    \ 

L4a;-13«/=78. 

fl8a;  +  5w  =  38, 
1 12  a;  — y=  —  5. 


15. 


16. 


17. 


18. 


19. 


20. 


21. 


22. 


23. 


24. 


r8x  +  9y  =  26, 
.  32  a;  -  3  y  =  26. 

33  a; +  54  2/ =  -24, 
.44a;-80«/  =  44. 

21a;-232/  =  2, 
.7a;-19y  =  12. 

15  a; +  28?/ =  157, 
20  a; +  21  ^  =  144. 

65a;+68t/  =  -3, 
39a;-119t/  =  158. 

63  a;  -  46  «/  =  29, 
42  a;  -  69 «/ =  96. 

27  a;  -  5  ?/  =  -  37, 
81a;- 7?/  = -151. 

13  a; -15?/ =  11, 

ll2a;-7y=17. 

11  a: +  13,?/ =  -9, 
15  a;—  14?/  =  —  44. 

19a;-23t,-  =  -ll, 

22  a; +25?/ =  -10. 


170  ELEMEXrAUY    ALGEBRA  [Cii.  XI,  §  1G(; 

II.   Elimination  by  Substitution 

156.    Solve|-^  +  ^^^  =  ^^'  ('-) 

I  -4  a: +  21  ^  =  55.  (2i 

Transposing  in  (1),  2x=13-3y,  (3) 

dividing  (3)  by  2,  x  =  ^^^~^^,  (4) 

substituting  x  from  (4)  in  (2), 

_4(^l^^/^  +  2l2/  =  55,  (5) 
simplifying  in  (5), 

^^^^^  +  21,  =  55,  (6) 
multiplying  (6)  by  2, 

-52  +  12^  +  422/  =  110,  (7) 

transposing  and  uniting  in  (7),       54  2/  =  162,  (8) 

dividing  (8)  by  54,                                 ?/  =  3.  (9) 

Substituting  y  from  (9)  in  (1),  2  a: +9  =  13,  (10) 

transposing  and  uniting  in  (10),       2  a;  =  4,  (11) 

dividing  (11)  by  2,                                a;  =  2.  (12) 

Verification:  4  +  9  =  13;   -8  +  63  =  55. 

It  is  to  be  noticed  that  the  above  equations  may  also  be 
solved  by  the  Addition  and  Subtraction  method. 

Rule  for  Elimination  by  Substitution:  In  one  of  the 
equations  find  the  value  of  one  ujiknown  quantity  in  terms 
of  the  othe7\  Substitute  the  value  thus  obtained  in  the  other 
equation.  Reduce  this  equation.  Verify  the  solution  iv 
each  of  the  given  equations. 


Cii.XI,§160]      SIMULTANEOUS   SIMPLE   EQUATIONS 


171 


EXERCISE   LXXVII 

Solve  the  following  systems  of  equations  by  substitution ; 


2. 


3. 


4. 


{^ 


8. 


9. 


10. 


11. 


12. 


2x  —  t/  =  0. 

-2x+t/  =  -B, 
-3a7  +  4«/  =  8. 

[Sx  +  z/=lS. 
r2a;  +  3«/  =  46, 

4x  +  ?/  =  23, 
3a;-2y  =  9. 

.2x  —  y  =■  15. 

42;  +  3«/  =  81, 
-2;+2«/  =  21. 

4a;+2«/=:38, 
3  a; -3  2/ =  6. 

f2a;  +  t/=20, 
l4a;+32/  =  70. 

p-9y  =  0, 
.4  a; -2/ =70. 

f4a;-5«/=3, 
l8a;  +  2«/  =  66. 

f  2  a;  -  2/  =  10, 
l3«/  +  17a;=177. 


13. 


14. 


15. 


16. 


17. 


18. 


19.    \ 


20. 


21. 


22. 


23. 


24. 


l7a;  +  32/  =  82. 

|3rc-6y  =  2, 
l4a;  +  72/  =  -93. 

J4a;+3y  =  4, 
l-7a;  +  5  2/  =  75. 

f  8  a;  —  5 «/  =  6, 
i7a;  +  10«/  =  149. 

r3a;  +  12«/  =  57, 

l2a;  +  ?/  =  10. 

r  7  a;  +  4  y  =  95, 
.a;— 2y=  — 7. 

r27a;  +  14^  =  41, 
1 36  a; +  51 3^=87. 

f  100  a; -143  2/ =  757, 
.llx-91«/  =  8. 

55^  +  31z/  =  171, 
27  a; -11 2/ =  18.4. 

r  109  a; +  110  2/ =  86, 
,107a;  +  146«^  =  98. 

83a;  +  25«/  =  4, 

21a;  +  85y  =  6. 

[39a;- 98^^  =  3, 
.51  a; +  182^  =  63. 


172  ELEMENTARY  ALGEBRA  [Ch.  XI,  §  157 

III.   Elimination  by  Comparison 

157.    Solve  i                     ^^  ^  ^ 

[    x+    2/ =  18.  (2) 

Transposing  in  (1),                   2  a;  =  16  —  3  ?/,  (o) 

dividing  (3)  by  2,                             x  =  ?^^^,  (4) 

transposing  in  (2),                            x  =  l^  —  y,  (5) 

comparing  x  in  (5)  and  (4),    18  —  t/  =  — i""^?  (^) 

multiplying  (6)  by  2,            36  -  2  2/  =  16  -  3  2/,  (7) 

transposing  and  uniting  in  (7),       ?/  =  —  20,  (8) 

substituting  ?/  in  (5),                         x  =  38.  (9) 
Verification:         76-60  =  16;  38-20  =  18. 

Rule  for  Elimination  by  Comparison:    In  each  equation 
find  the  value  of  one  unknown  in  terms  of  the  other.     Place 

these  values  equals  and  solve  the  resulting  equation.  Verify 
the  solution  in  each  of  the  given  equations. 


exercise  lxxviii 

Solve  the  following  systems  of  equations  by  comparison, 
and  check  the  results  on  the  graph : 

^     r5^+^=7,  ^    |:^,+  4^  =  7, 


2     '^•f2/  =  0, 
2x+?^y  =  l. 


'^+3?/  =  3, 
^x+^y  =  -l. 


2x-\-^y  =  10,  ^3^+2y  =  0, 

6. 


rAx 

[?>x 


+  22/  =  10.  [2x-y=--l 


Ch.X1,§158]      simultaneous   SIMPLE   EQUATIONS  173 

158.  If  either,  or  both,  of  the  equations  in  a  system  of 
equations  contain  aggregations  or  fractions,  it  is,  in  gen- 
eral, best  to  simplify  the  equations  before  elimination. 


Solve 


4(:^;-3^/)  =  8,  (1) 

^^  =  3.  (2) 


Sim  plif yiiig  in  (1),                    4  a;  - 12  ?/  =  8,  (3) 

multiplying  (2)  by  i»  —  2  y,                  a;  +  ?/  =  3  a;  -  6  2/,  (4) 

transposing  and  uniting  in  (4),  —  2  a;  +  7  ?/  =  0,  (5) 

multiplying  (5)  by  2,                —  4  ic  + 14  ?/  =  0,  (6) 

rewriting  (3),                                  4  a;  -  12  y/  =  8,  (7) 

adding  (6)  and  (7),                                    ^  2/  =  8,  (8) 

dividing  (8)  by  2,                                         ^  =  4.  (9) 

Substituting  y  from  (9)  in  (3),  4  a;  -  48  =  8,  (10) 

transposing  and  uniting  in  (10),             4  a;  =  66,  (11) 

dividing  (11)  by  4,                                       a;  =  14.  (12) 

14  +  4     o 


Verification  :   4(14  — 12)  =  8 ; 


14-8 


EXERCISE    LXXIX 

Solve  the  following  systems  of  equations,  selecting  the 
best  method  : 

i2(5rr-y)-32/  =  5. 


174 


ELEMENTARY   ALGEBRA 


[Ch.  XI,  §  158 


f      7      _     1    ^ 
3.    l2x  —  t/     x  —  y'' 

g     |5:r-(3./-i)=f, 


'^±1=6, 
x-y 

x  +  1  ^1 

x  +  ly      3* 

6a;  — y     _  1 


8.    ^ 


6. 


\i)  +  x      10  +  ?/ 
17  3 


=  0, 


9. 


Ix—by     x+S 


5(a;  +  2  2/)     29 
5a;  +  y^2 

f£+j_+4^g^ 
X  — Z/  +  6 
2a;-.y  +  7_      ^ 
a;-2«/  +  7 


10.    J 


11. 


'     8a;-3,y     _^ 
5x-2«/  +  3 
4a;  +  2y  +  ll^      4 
6x— 7i/  +  6  3 

3.y-Tx     io_a;-.y     .y. 
4  8        3 


12.    . 


13.    J 


a^  +  1  1  y  +  2^2C.y-a;)^ 

4  10              5 

a;-l  ,y-2^3.y-8a;. 

4  12             18      * 

f4a;  +  .y-4  6a;  +  2y-7_» 

3  9 

2a:  —  y  +  l  10  a;  —  4y_j^ 
8  3 


Ch.XI,§]68]       simultaneous  SIMPLE  EQUATIONS  175 


14.    -! 


^x  +  y  +  o      2x-y  +  5  ^x  +  2i/  +  l 
9  6  2         ' 

2x  —  t/-\-7  _  4x-'dt/-l  _  5x  +  8y  +  S 
8  4  ~  16         * 


15    |(3^+8)(4^-3)  =  (2:r  +  9)(6y-5), 
l(2x-l)(12y-l)  =  (3a:  +  8)(8.y-7). 


16. 


f7.v      5.y  +  22_a;      55-8.y 
10  7  5  6' 

— ! — ^  =  —rX  +  zy—  19. 

7  11  11  "^ 

f2x  +  7.y  +  5  Q^_4x+ll.y  +  5^ 


17.  ^ 


18. 


o        5a;+3.y_llrc-14,y+241 

^"^      ir"~       154 

o      ,  4,y+5a:  ,  9g+8.y-12_  1    ,  lla:+6.y4-l 
SX+         ^—+  -  -+  - 

7  12  8 


19. 


X  — 


x  + 


2y  —  X 

22 
x+1 


20- 


49-2a; 


9  6 


r 

20.    J 


3     "^     4  6    ' 

2y+7      3a;-y_3.y— 2a;4-4 

8  7      ~  8  * 


21. 


3.y-2  _  x-5  ^  r  _  2a;+3.y-l^ 

4  2  8  ' 

5a;+6«/-3     2^+9w-2_      „ 


176  ELEMENTARY  ALGEBRA  [Ch.  XI,  §  159 

159.    It  is  often  convenient  in  simultaneous  equations 
containing  fractions  to  eliminate  one  of  the  fractions. 


1 .    Solve 


■^  _  2/  _  1 
2      i~    ' 

U      12"      ''• 


4     12 
Multiply  (1)  by  i,  1-1  =  1' 


subtracting  (3)  from  (2), 

12^8  2' 

multiplying  (4)  by  24, 

-102/  +  37/  =  -72--12, 
uniting  in  (5),  —  7  ^  =  —  84, 

dividing  (6)  by  —7,  *    2/  =  12. 

Substituting  2/  from  (7)  in  (2), 

|-5  =  -3, 

transposing  and  uniting  in  (8),   t  =  2, 

multiplying  (9)  by  4,  x  =  8. 

.J  ,  '8     12     -,      8     60         o 

Verification:  - — —==1'^    -——=  —  3. 

^4:  4:  liiJ 


This  method  is  especially  valuable  in  solving,  by  the 
foregoing  methods,  equations  which  conta^'a  the  unknowns 
in  the  denominators. 


Ch.XI,§159]      simultaneous  SIMPLE  EQUATIONS  177 


2.    Solve 


^-  +  1=5,  .  (1) 

1^-^  =  2.  ••  (2^ 

X       y 

16_28^2 

X       y 

Multiplying  (1)  by  7,    ?i  +  —  =  36,  (3) 

X       y 

adding  (2)  and  (3),  —  =  37,  (4) 

multiplying  (4)  by  x,  37  =  37  a;,  (6) 

dividing  (5)  by  37,  x  =  l.  (6) 

Substituting  x  from  (6)  in  (1), 

3  +  ^  =  5,  (7) 

y 

transposing  and  uniting  in  (7), 

i-2,  (8) 

multiplying  (8)  by  ?/,  4  =  2  y,  (9) 

dividing  (9)  by  2,  2/  =  2.  (10) 

Verification:  ?  +  ^  =  5;  ^-?5  =  2 

Although  equations  (1)  and  (2)  can  be  solved  by  first 
multiplying  each  equation  by  xy^  and  the.n  multiplying 
the  resulting  equations  by  2  and  5  respectively  and  next 
3ubtracting  these  last  equivalent  equations,  this  method  is 
not  recommended.  If  equations  are  solved  by  the  latter 
method,  it  may  happen  that  roots  are  introduced  which  do 
not  verify. 


178 


ELEMENTARY   ALGEBRA 


[Ch.  XI,  §  159 


EXERCISE    LXXX 


Solve  the  following  systems  of  equations  by  eliminating 
the  fractions : 


2. 


3. 


4.     \ 


5. 


6. 


^1+1=7, 

f?+5  =  6, 

6     5 

7.     ' 

X     y 

^-4=7. 

M_21^3. 

3     16 

I  a;       y 

|  +  ?=11, 

ff +  ''  =  18, 

3     4 

8. 

4iX      by 

1  +  1  =  5. 

r+o'  =  12. 

l7     8 

l3a;     9^/ 

'2-f  =  6, 

r  3    ,  5_11 

6     4       ' 

9. 

2a;      y       4 

f-|  =  4. 

.'+o'=l- 

,7     2 

13a;      9y 

ff  +  '^/  =  17, 

[f      /=7, 

5       8 

10. 

7x     by 

?     ^2^=     7. 
13       4 

11      13      19 

16a;      by      3* 

x  ,  w     63 

fo     ,8      44 

3a;  +  -^=  — , 

9     7      10 

11. 

by      3- 

X     52,y     392 

a;       1       4 

I  3  '    56        10 

l4      ^y     9 

\ll     '^/  =  16, 

^i+i=9, 

10       7 

12. 

X      «/ 

5?  ^y=\Q 

X     8 

I  8       35 

l^^     7' 

Ch.XI,§160]      8IMULTANE07JS   SIMPLE    EQUATIONS  179 

160.  Literal  simultaneous  equations  are  solved  in  the 
same  way  as  are  numerical  equations.  Especial  care 
should  be  taken  to  express  the  values  of  the  unknowns 
in  terms  of  the  knowns;  and  to  that  end  the  known  terms 
should  always  be  transposed  to  the  right  member  of  the 
(equation.    ' 

1.  Solve  P  +  r'?  ^'^ 

Transposing  in  (2),  4:X  +  y  =  5  a,  (3) 

rewriting  (1),  x  +  y  =  2  a,  (4) 

subtracting  (4)  from  (3)>  3  a;        =  3  a,  (5) 

dividing  (5)  by  3,  x  =  a.  (6) 

Substituting  x  from  (6)  in  (1),      a  +  y  =  2  a,  (7) 

transposing  and  uniting  in  (7),  y  =  a.  (8) 

Verification  :  a  +  a  =  2  a;  a  =  5a— 4  a. 

2.  Solve   ,«-  +  ^^  =  *^'  (1) 


_bx  +  ay  =  h^.  (2) 

Multiplying  (1)  by  h,  abx  +  %  =  a%  (3) 

multiplying  (2)  by  a,  ahx  +  ary  =  ab^,  (4) 

subtracting  (4)  from  (3),  b^y  —  ary  =  a'b  —  aW,  (5) 

factoring  in  (5),  y  (b^  —  a?)  =  ab  {a?  —  W),  (6) 

dividing  (6)  by  W-o?,  y  =  -  ab.  (7) 

Substituting  y  in  (1),  ax  —  ab^  =  a%  (8) 

dividing  (8)  by  a,  x—b^  =  a\  (9) 

ti-ansposing  in  (9),  x  =  a^  +  b^  (10) 

Verification  : 

a(a'  +  b')  +  b(-ab)  =  a^',  b(a' +b') +a(-ab)  =  b\ 


180 


3.    Solve  < 


^A  +  -l  =  62  +  c2, 
ax      by 


.  hx      cy 


Multiplying  (1)  by  | 
multiplying  (2)  by  -, 


subtracting  (4)  from  (3), 


r   ALGEBRA               [Ch 

9 

XI, 

§160 

(1) 
(2) 

X     Q-y               0 

(3) 

1       W  _^-i^     W 
X     acy              a ' 

(4) 

ac       W  _ac^     b^ 
b^y     acy      b       a' 

(5) 

multiplying  (5)  by  ab^cy,  (j^(?  +  6*  =  Q?-b(?y  +  b^cy^       (6) 

factoring  the  right  member  in  (6),  aV  +  ?>'*  =  bey  (aV  +  6^),      (7) 
dividing  (7)  by  (aV  +  b%  1  =  bey,  (8) 


dividing  (8)  by  5c, 

^      be 

(9) 

Substituting  y  from  (9)  in  (1), 

ax      b               ' 
be 

(10) 

simplifying  in  (10), 

A  +  c^==.^  +  c^, 

(11) 

transposing  and  uniting  in  (11), 

ax        ' 

(12) 

dividing  (12)  by  6, 

ax       ' 

(13) 

multiplying  (13)  by  ax^ 

1  =  abx, 

(14) 

dividing  (14)  by  a&, 

1 

X=-T' 

n.n 

(15) 

Verification  :   -  +  ^  =  ?y^  +  c^ ;  -  —  -  =  a^  - 


Cii,XI,§lCO]      SIMULTANEOUS   SIMPLE   EQUATIONS 


181 


EXERCISE  LXXXI 


Solve  the  following  systems  of  equations  and  verify  the 
results : 


2. 


3. 


5. 


7.    i 


8. 


9. 


10. 


'  x+  ai/  =  a% 
X—  hy  =  h\ 

a;  +  hy  =  1)^, 

ax+hy  =  (?, 

X     m 

'  a(x+y)  +  h(x-y^  =  c, 
X  _m 
^y"  n 

ax+hy  =  e^ 
a^x  +  ^1  =  c^y. 

ax+ly  =  2, 
ab(x+  y^=  a  +  h. 

{ax  =  h{y-2), 
a2+ J2 


y-x-- 


ah 


11. 


12. 


13. 


'  a(x+c)+h(]j—c)=a?—b\ 
y—x=2c. 

J2_^2 


14. 


x  —  {a+  V)y= 

(h  —  a)x  +  ahy  =  V^, 

a      h      cfi—V^ 
.  (a+V)x+(a—K)y=:^a+K 

ax  .  hy  ,7 

6        a 

X  y  __a^  +h^ 

a  b         c?}?" 


15. 


a+b 

bx+  a2 


a5, 


=  1  + 


a^/  =  ^(a;  +  1)  —  a, 

(a+6+^X^-a) 

a^n;  —  b'^y  =  a  +  J, 
5:z;  —  a^/  =  —  1 . 

a:r  —  (a  —  b^y=  (a—  6)2, 
bx—y  =  b(a  —  b—V). 


16. 


2/—  6      a—  c 

X  —  b  __a  +  c 

^y  —  a      b  —  c 

y 


17.    \ 


b  -\-  c      a  +  € 


a  —  C'^ 


.-,-^  =  5. 


a-\-  b      b  +  c 


182  ELEMENTAUY   ALGEBRA  [Cii.  XI,  §  IGl 

THREE   OR   MORE   UNKNOWN   QUANTITIES 

161.  Three  simultaneous  equations  containing  three 
unknowns  are  solved  by  the  elimination  of  one  of  the 
unknowns  between  a  pair  of  the  given  equations,  and  by 
the  further  elimination  of  the  same  unknown  between  a 
different  pair  of  the  given  equations ;  the  resulting  equa- 
tions are  then  solved  as  in  §§  155-7. 

Elimination  is  performed  by  the  addition  and  subtrac- 
tion method.  That  quantity  is  generally  chosen  for  elimi- 
nation whose  coefficients  are  smallest.  It  is  evident  that 
of  three  given  equations  the  first  may  be  combined  with 
the  second,  the  first  with  the  third,  and  the  second  with 
the  thirde 

'x  +  7/  +  Z=U,  (1) 

1.    Solve     i4:x  +  27/  +  z=:4:S,  (2) 

[Qx  +  5y  +  z=88.  (3) 

Subtracting  (2)  from  (1),  -.3x-y  =  -29,  (4) 

subtracting  (3)  from  (2),        —  5  cc  —  2/  =  —  45,  (5) 

subtracting  (5)  from  (4),                   2x  =  16,  (6) 

dividing  (6)  by  2,                                 a;  =  8.  (7)  ^ 

Substituting  x  from  (7)  in  (5), 

_.  40-2/ =  -45,  (8) 

transposing  and  uniting  in  (8),           2/  =  ^-  (^) 

Substituting  x  from  (7)  and  y  from  (9)  in  (1), 

8  +  5  +  ^  =  14,  (10) 

transposing  and  uniting  in  (10),         z  =  l.  (11) 

Verification  : 

84-5  +  1  =  14;  32  +  10  +  1  =  43;  72  +  154-1=88. 


Ch.XT,  §102]      SIMULTANEOUS   SIMPLE   EQUATIONS  183 

162.  Four  or  more  simultaneous  equations  containing 
four  or  more  unknowns  are  solved  by  the  elimination  of 
one  of  the  unknowns  between  three  or  more  pairs  of  the 
given  equations,  in  the  resulting  equations  another  un- 
known is  eliminated  between  two  or  more  pairs  of  the 
resulting  equations,  and  the  process  is  continued  until 
three  resulting  equations  are  obtained.  These  latter  equa- 
tions are  solved  by  the  method  shown  in  §  161. 

Care  must  be  taken  to  keep  the  same  number  of  equa- 
tions as  unknowns ;  otherwise,  dependent  equations  will 
be  obtained. 

x  +  y  +  z  +  w  =  -ig^  (1) 

2x  +  y  +  ^z  +  2tv  =  l,  (2) 

^x  +  2y  +  2z  +  ^'W=:^^,  (3) 

4a:+3J/  +  4^  +  6^^  =  \^.  (4) 


1.    Solve 


Eliminate  y 

Subtracting  (2)  from  (1),     —x  —  2z  —  io=:^  —  ^-,  (5) 

subtracting  (3)  from  (1)  x  2,  —  a;        —  6  ic  =  —  |,  (6) 

subtracting  (4)  from  (1)  x  3,  —x  —  z  —  ^w^^  —  ^-^.  (7) 

Eliminate  z 
Subtracting  (7)  x  2  from  (5),       a?  +  5  lo  =  -i/,  (8) 

rewriting  (6),  —  oj  —  6  2^  =  —  J.  (6) 

'    Eliminate  x 
'  Adding  (8)  and  (6),  ~     ^w  =  -\,  (9) 

dividing  (9)  by  -  1,  w  =  \.  (10) 

Whence,  by  substitution, 

a;=l,  2/  =  i  z  =  \,w^. 


184 


ELEMENTARY   ALGEBRA 


[Cii.  XI,  §  16^ 


EXERCISE   LXXXII 

Solve  the  following  systems  of  equations : 


1. 


'4:X 

+  5?/  +  92  = 

13, 

5x 

+    y+2z  = 

-5, 

Ix 

-5^-82  = 

-31 

2. 


3. 


f    x+    5y  —  2z  =  5^ 

I  Sx-\-    8?/  +  4^  =  31, 
[1  x+2oy-4:z  =  4:5. 

2x-9y+10z  =  55, 
llx—'dy—    5z=7^ 
ISx  —  iy—    62^  =  1. 


4. 


z  —  -,  t) 


5. 


'5x+Si/—2z  =  %, 
4  x 

(Sx-^-l/-  ^-12 
^i2;+Jy+|3  =  14, 

s^J  +  i?/- ^2  =  5, 
i^+    y-j2  =  12. 


la;+32/-fi 


16, 


6.  i  2x-y  +  lz  =  25, 
'  lx-iy+    s  =  17|. 


I  3 

'1-1.2, 

X      y 

1-1  =  8, 

y    2 

1  +  1  =  9. 

_  Z        X 


10. 


fl 

-  + 

X 

1_ 

=  7, 

- 

1 

-  + 
y 

2_ 
2"" 

=  14, 

X 

8_ 

2 

21. 

fl 

-  + 

X 

3 

-  + 

y 

4_ 

2 

8, 

4 

x-^ 

5 

2_ 

2 

16, 

7 

X 

y 

4_ 

21. 

f3 

-  + 

X 

4 

8_ 

2~ 

15, 

6 
a; 

1 

2y 

2 

+  - 
2 

=  h 

9 

[4a; 

8 

+  - 

y 

+1. 

2 

=  13. 

x  +  y-- 

=  2« 

', 

ay  +  z-. 

=  a2. 

hx 

—  Z  : 

=  52. 

11. 


{ax  +  y=^l, 

12.  -!  hx-\-z  =  l, 

I  cz  +  X  =  bo. 
bx+ay=2  ab, 

13.  ^  cy+bz  =  2  bc^ 

^  ex+  az  =  2  ac. 


Ch.XI,  §162]       SIMULTANEOUS  SIMPLE   EQUATIONS 


185 


15. 


16. 


17.   i 


x+  ay  =^a(a-\-h^^ 
14.  \  a^z  —  bx  =  a^, 
y  =  z-a. 

'  ax  +  y  -\-  z  =  ahc  +  a(b-\- 1?), 

-  x-{-hy  -\-  z  =  ahc  +  b(a  +  c?), 

x+  y  -{-  ez  =  ahc  +  c{a  +  6). 

x-\-    y  +     z  +  u  =  65^ 
x+2y  —     z  —  u  =  l^ 
2x+3y-j-2z-u=m, 

^Sx—2y-{-2z  +  ii  =  54:. 

'  x+2y-{-  z—  21  =  10, 
X—  y  +  2z  +  u  =  23, 
x+Sy  +  4:z-2u  =  89, 
X—  5y  —  4:Z—  8u  =  41. 

^x+      y+      z-\-      u  =  10, 
x+    8y+    5z+    lu  =  SO, 

x+    6^+15^  +  28^^  =  80, 
^  +  10  2/  +  35  ^  +  84  ^  =  188 


'  {b+  c)x+by  =  c^ 

(a  -\-  c)y  +  cz  =  a, 
(^a  +  b)z  +  ax  =  bo 


18. 


19. 


20. 


21. 


'a;+5^=23, 
y  +  4z  =  -l, 
z+Su=  20, 

u-^2v  =  3. 
V  +  x  =  6. 


\y  +  lz- ^21  =  16. 

2x+3y  =  57, 
5x-iz  =  20, 
22.  -!  32+2^  =  48, 
4y+3v  =  68, 
1  u  —  6v=^  15. 


186 


ELEMENTARY   ALGEBRA 


[Ch.XT,§162 


REVIEW   EXERCISE  LXXXIII 

Solve  the  following  simultaneous  equations: 


2. 


4.    ^ 


'  ax  +  by  = 

1, 

hx—  ay  = 

1. 

r  ax  =  Jy, 

hx  +  ay  = 

e. 

X 

15 

4' 

,x      2 

'2      3      , 
-  +  -  =  4, 
X     y 

1+1  =  6. 

[X     y 

6. 


%x 


2y-5_ 


4-9a; 


^l 


a;-2  3 

lx-3y=10. 

x  +  y  =  2a, 
[  (a  -  h)x  =  (a  +  h')y. 

— +^  =  e, 


1 


=  (i. 


2x-Z    y-2  ,        7 

2x—y     '2y—x_  _49 
3  4      "     12* 


9. 


10. 


hx      ay 

'3^  +  2^-42=15, 
5^-3^+255=28, 
.3?/  +  42  —  a;=24. 

+  -^  =  2a, 


11. 


12. 


13. 


5(rr-2i/)-(a;-2/)=-24, 
[ll(2:z;+32/)+(2:^-2/)=200. 

qx  —  rb  =  p(a  —  ?/), 


ql+r  =  p(i  +  ^ 


(^D-K^D=*. 


a+  6      a  —  6 


Oh.  XT,  §1(52]      SIMlTLTANi:0lJ8   SIMPLE   EQUATIONS 


187 


14. 


15. 


16. 


(a  -  b)x  -  (^  +  h)7/  =  2  a2  -  2  b% 
(a  +  b)x  —  {a  —  b)y  =  4  ab. 

ax—by  =  €?  —5^—2  a6, 
bx  +  ay  =  2ab  +  a?'—  b\ 

y^=b^a. 


a  +  b      b+  c 

y    ,    ^ 

c+  a 

z 


c—  a 

X 


=  b- 


17.    \ 


b  —  c      a—  b 

r  {a  +  />)  (a;  +  2/)  -  (^  -  ^)  (^  -  ^)  =  «^ 
(a  -  />)(:?;  +  ^)  +  («  +  ^)(^  -  2/)  =  ^*^- 

+  ~-^  =  2a, 


18.    ^ 


^  —  y_  ^  +  y , 


19.    ^ 


20. 


I  2a^>       a2+ J2 

a;  +  y  -  ^  =  7, 
y  +  2  -  w  =  9, 
z  +  u—.x=19^ 
.u+  X  —  y  =  13. 

^  +  2/  +  ^  =  0, 

(^  +  h)x  +  (a+  c)y  +  (b  +  a)z  =  0, 

Qc  —  b)x  +(a  —  c)y  +  (J  -  a^z  =  2(a J  +  «(?  +  bd) 
-2(a2+62+^2), 


CHAPTER  XII 

PROBLEMS   INVOLVING   SIMPLE  EQUATIONS 
EXAMPLES 

163.  1.  The  sum  of  two  numbers  is  27,  and  if  the  greater 
be  divided  by  the  less,  the  quotient  is  1  and  the  remainder 
is  5.     Find  the  numbers. 

Let  X  =  the  greater  number,  and  y  =  the  less  number. 

By  the  first  condition, 

by  the  second  condition, 

Solving  (1)  and  (2), 

Verification:  .^  , _,  , 

It  should  be  noticed  that  in  this,  as  in  many  of  the  fol- 
lowing problems,  one,  two,  or  more  unknowns  may  be 
employed  to  find  the  solution. 

Let  X  =  the  greater  number,  and  27  —  a?  =  the  less  number. 
By  the  second  condition,     —^ — =  1.  (1) 

Solving  (1),  x  =  16  2ind  27  -x  =  11. 

In  general,  if  an  equation  can  be  solved  with  a  single  un- 
known, this  method  is  preferable. 

188 


x  +  y  =  27, 

(1) 

a;  — 5_-. 

y 

(2) 

a;  =  16  and  y  =  ll. 

5  +  11  =  27:^^7^  = 

=  1. 

Ch.  XII,  §  163]  SIMPLE   EQUATIONS  189 

2.  The  width  of  a  rectanguhir  room  is  f  of  its  length. 
If  the  width  were  5  feet  more,  the  room  wotdd  be  square. 

Find  the  dimensions. 

5  X 
Let  X  =  number  of  feet  in  the  length,  and  -—  =  number  of 

feet  in  the  width. 

By  the  conditions,  — ■  +  5  =  x,  (1) 

Solving  (1),  a;  =  30 ;  hence  ^  =  25. 

Verification  :  25  + o  =  30. 

3.  A's  age  is  ^  of  B's  age,  but  5  years  ago  A  was  ^  as 
old  as  B.     Find  their  present  ages. 

Let  X  =  the  number  of  years  in  A's  age,  and  5  x  =  the 
number  of  years  in  B's  age. 

By  the  conditions,  9(x  —  5)  =  5x  —  5.  (1) 

Solving  (1),  a;  =  10 ;  hence  5x  =  50, 

4.  A  can  row  4  miles  an  hour  down  a  stream,  and  2 
miles  an  hour  against  the  stream.  Find  A's  rate  in  still 
water,  and  the  rate  of  the  stream. 

Let  X  =  A's  rate  in  still  water,  in  miles  per  hour ;  and  y  = 
rate  of  stream,  in  miles  per  hour. 

By  the  first  condition,         x  +  y  =  4:]  (1) 

Dy  the  second  condition,         x  —  y  =  2.  (2) 

Solving  (1)  and  (2),         x=3,  y  =  l. 

5.  At  what  time  between  2  and  3  will  the  hands  of 
a  clock  be  («)  together  ?  (J)  exactly  opposite  ? 

In  the  same  period  of  time  the  minute  hand  moves  twelve 
times  as  fast  as  the  hour  hand.  Thus,  the  minute  and  hour 
hand  cover  in  an  hour  respectively  60  and  5  minute-spaces; 
and  in  12  minutes  respectively  12  and  1  minute-spaces. 


190 


ELEMENTARY  ALGEBRA 


LCh.  XII,  §  163 


Let  X  =  number  of  minute-spaces  passed  over  by  the  minute 
hand  in  given  time,  and  —  =  number  of  minute-spaces  passed 
over  by  the  hour  hand  in  given  time. 


(a)  Since  the  minute  hand  starts 
at  XII  and  moves  to  A,  where  it 
meets  the  hour  hand  which  starts 
from  II,  which  is  10  minute-spaces 
from  XII,  and  in  the  same  time 
moves  to  A,    by  the  conditions, 


aj  =  10  + 


12 


Fig.  9. 


Solving  (1),  X  =  10|^. 


(1) 


(b)  Since  the  minute  hand 
starts  at  XII  and  moves  to  B, 
where  it  is  exactly  opposite  the 
hour  hand,  which  starts  from 
II,  10  minute-spaces  from  XII, 
and  in  the  same  time  moves  to 
Ay  by  the  conditions, 


^=10  +  j|+30. 
Solving  (1),  X  =  43^p 


(1) 


Fig.  10. 


6.  The  sum  of  the  two  digits  of  a  number  is  6,  and 
if  36  be  added  to  the  number  the  order  of  the  digits  is 
reversed.     Find  the  number. 

Since  in  arithmetic,  position  indicates  the  value  of  the  digits 
in  a  number,  (56  =1 10  •  5  +  6),  let 


Cii.  XII,  §  163]  SIMPLE   EQUATIONS  191 

X  =  the  digit  in  the  tens'  place, 
and  y  =  digit  in  the  units'  place, 

and  10  x-{-y=  the  number.  . 

By  the  first  conditions,  aj  +  ?/  =  6,  (1) 

Dy  the  second  condition,        10  x  +  y +  36  =  10  y  -{-x.  (2) 

Solving  (1)  and  (2),  x  =  1,  y  =  o]  hence  the  number  is  15. 

7.  A  can  do  a  piece  of  work  in  5  days,  and  with  the 
help  of  B  can  do  it  in  3  days.  How  long  would  it  take 
B  alone  to  do  the  woi*k  ? 

Let  X  =  the  number  of  days  it  takes  B  alone  to  do  the  work, 
then    -  =  part  that  B  can  do  in  1  day, 

X 

and      -  =  part  that  A  can  do  in  1  day, 

and      -  =  part  that  A  and  B  can  do  in  1  day. 

By  the  conditions,  _  _|.  _  =  _ .  (1) 

O       X       o 

Solving  (1),  X  =  71 

8.  A  train  runs  84  miles  in  the  same  time  that  a  second 
train  runs  96  miles.  If  the  rate  of  the  first  train  is  3  miles 
per  hour  less  than  that  of  the  second  train,  find  the  rate  of 
each. 

Let  a;  =  rate  of  the  first  train,  and  x  +  3  =  rate  of  the  second 
train. 

By  the  conditions,  —  =  -^^'  (1) 

X         X  -f-  o 

Solving  (1),     07  •■=  21 ;  hence  x-{-3  =  24. 


192  ELEMENTARY  ALGEBRA  [Ch.  XII,  §  163 

9.  A  number  of  4  ^o  bonds  were  sold  at  90,  and  the  pro- 
ceeds invested  in  3J  ^  bonds  at  75,  the  par  value  of  each 
bond  being  $100.  If  the  gain  in  income  is  14,  find  the 
number  of  4  ^  bonds. 

Let  X  —  the  number  of  4  %  bonds, 

then  4  a;  =  the  income  in  dollars  of  the  4  %  bonds, 

and  90  cc  =  the  value  in  dollars  of  the  4  %  bonds, 

90  X 
then  — —  =  the  number  of  3|^  %  bonds, 

/90  x\ 
and    Z\  [ ]  =  the  income  in  dollars  from  the  ?>\  %  bonds. 

By  the  conditions,         3|  (^^^-^  -  4  a;  =  4.  (1) 

Solving  (1),  a;  =  20. 

EXERCISE   LXXXIV 

1.  The  sum  of  half  a  number  and  its  third  part  is  135. 
Find  the  number. 

2.  The  difference  between  the  third  and  seventh  parts 
of  a  number  is  40.     Find  the  number. 

3.  The  excess  of  the  sum  of  the  fourth  and  twelfth 
parts  over  the  ninth  part  of  a  number  is  8.  Find  the 
number.     ' 

4.  The  excess  of  the  sum  of  the  fifth  and  seventh  parts 
over  the  difference  of  the  half  and  the  third  parts  of  a 
number  is  259.     Find  the  number. 

5.  Find  that  number  which  is  1^  times  the  excess  of  the 
number  over  2. 

6.  The  sum  of  two  numbers  is  32,  and  their  difference 
is  8.     Find  the  numbers. 


|(Jh.  XII,  §  163]  SIMPLE   EQUATIONS  193 

7.  Tlie  difference  of  two  numbers  is  13,  and  if  144  be 
subtracted  from  8  times  the  first,  the  remainder  is  56. 
Find  the  numbers. 

8.  The  fourth  part  of  the  larger  of  two  consecutive 
numbers  exceeds  the  fifth  part  of  the  smaller  by  1.  Find 
the  numbers. 

9.  The  sum  of  two  numbers  is  18,  and  if  the  greater 
number  be  divided  by  the  less,  the  quotient  is  2.  Find 
the  numbers. 

10.  Find  the  two  numbers  such  that  their  difference  is 
20,  and  the  quotient  of  the  greater  divided  by  the  less  is  3. 

11.  The  sum  of  two  numbers  is  26,  and  if  the  greater 
number  be  divided  by  the  less,  the  quotient  is  1  and  the 
remainder  is  4.     Find  the  numbers. 

12.  The  difference  of  two  numbers  is  9,  and  if  the 
greater  be  divided  by  the  less,  the  quotient  is  2  and  the 
remainder  is  2.     Find  the  numbers. 

13.  The  difference  of  two  numbers  is  18,  and  if  the  less 
be  divided  by  the  greater,  the  quotient  is  ^.  Find  the 
numbers. 

14.  The  sum  of  two  numbers  is  22,  and  if  the  less  be 
divided  by  the  greater  diminished  by  7,  the  quotient  is  -|-. 
Find  the  numbers. 

15.  The  sum  of  two  numbers  is  200,  and  their  difference 
is  I  of  the  less  number.     Find  the  numbers. 

16.  The  sum  of  two  numbers  is  59,  and  if  the  greater 
be  divided  by  the  less,  the  quotient  and  the  remainder  is 
4.     Find  the  numbers. 


194  ELEMENTARY   ALGEBRA  [Ch.  XII,  §  163 

17.  The  difference  of  two  numbers  is  16,  and  if  the 
g'reater  be  divided  by  the  less,  the  quotient  is  2  and  the 
remainder  is  4.     Find  the  numbers. 

18.  If  59  be  added  to  half  of  a  certain  number,  the  sum 
obtained  is  1^^  times  a  seventh  of  the  number.  Find  the 
number. 

19.  A  number  is  10  times  a  second  number.  The  quo- 
tient of  the  first  number  divided  by  22  exceeds  by  -^^  the 
quotient  of  the  second  number  divided*  by  3.  Find  the 
numbers. 

20.  If  a  certain  number  be  added  to  the  terms  of  |,  it 
becomes  |.     Find  the  number. 


21.  Find  the  fraction  such  that  if  1  be  added  to  the 
numerator  it  becomes  ^ ;  but  if  1  be  subtracted  from  the 
denominator  it  becomes  \, 

22.  Find  the  fraction  such  that  if  3  be  added  to  the 
numerator  it  becomes  | ;  but  if  1  be  subtracted  from  the 
denominator  it  becomes  ^. 

23.  Find  the  fraction  sjach  that  if  4  be  subtracted  from 
its  terms  it  becomes  J  ;  but  if  5  be  added  to  its  terms  it 
becomes  |. 

24.  The  sum  of  two  fractions  whose  numerators  are 
respectively  7  and  9  is  '^-^-^-;  but  if  the  numerators  be 
interchanged,  the  sum  of  the  fractions  is  ^f .  Find  the 
fractions. 

25.  A  certain  fraction  becomes  y^g  ^^  ^  ^®  subtracted 
from  the  numerator,  and  becomes  ^  if  4  be  added  to  tlie 
denominator.     Find  the  fraction. 


Ch.  XII,  §  163J  SIMPLE  p:quations  195 

26.  If  3  be  added  to  the  numerator  and  1  be  added  to 
the  denominator  of  a  certain  fraction,  it  becomes  |  ;  but  if 
1  be  subtracted  from  the  numerator  and  3  be  subtracted 
from  the  denominator,  it  becomes  ^-.     Find  the  fraction. 

27.  The  sum  of  two  fractions  whose  numerators  are  each 
1  is  ^|-.  The  first  fraction  exceeds  the  second  by  J-^,  Find 
the  fractions. 

28.  The  width  of  a  rectangular  room  is  |  of  its  length. 
If  the  wddth  were  3  feet  more,  the  room  would  be  square 
Find  the  dimensions  of  the  room. 

29.  The  dimensions  of  a  rectangle  are  respectively  12 
feet  more  and  8  feet  less  than  the  side  of  an  equivalent 
square.     Find  the  dimensions  of  the  rectangle. 

30.  The  length  of  a  rectangular  floor  exceeds  the  width 
by  6  feet.  If  the  width  be  increased  by  3  feet  and  the 
length  b}^  2  feet,  the  area  is  increased  by  134  square  feet. 
Find  the  area. 

31.  A  square  contains  the  same  area  as  a  rectangle 
whose  dimensions  are  respectively  the  half  and  the  double 
of  the  side  of  the  square.  If  the  width  of  the  rectangle 
be  increased  by  3  feet  and  its  length  be  diminished  by 
5  feet,  the  area  is  increased  34  square  feet.  Find  the  side 
of  the  square. 

32.  Seven  men  and  5  boys  earn  $11.25  per  day,  and  at 
the  same  wages  12  boys  and  4  men  earn  $11  per  day. 
What  are  the  wages  per  day  of  a  man  ? 

33.  A  sum  of  money  is  divided  equally  among  a  certain 
number  of  men.  If  there  were  4  more  men,  each  would 
receive  $1  less;  if  5  less  men,  each  would  receive  $2 
more.     Find  the  number  of  men. 


« 


196  ELEMENTARY   ALGEBRA  [Ch.  XII,  §  163 

34.  A  could  have  bought  5  more  oranges,  each  at  half 
a  cent  less,  for  the  same  amount  of  money  that  he  could 
have  bought  3  less  oranges,  each  at  half  a  cent  more. 
Find  the  cost  of  the  oranges. 

35.  A's  age  is  ^  of  B's.  Five  years  ago  A  was  ^  as  old 
as  B.     Find  their  present  ages. 

36.  A's  age  is  five  times  B's.  In  12  years  B's  age  will 
be  ^  of  A's.     Find  their  present  ages. 

37.  A  is  50  years  old,  and  B  is  25.  In  how  many  years 
will  B  be  ^2  ^s  old  as  A  ? 

38.  A's  age  is  twice  that  of  his  son,  but  10  years  ago  it 
was  three  times  as  great.     Find  the  present  age  of  each. 

39.  If  A  was  four  times  as  old  as  B  7  years  ago,  and  if 
A  will  be  twice  as  old  as  B  in  7  years,  what  is  the  present 
age  of  each  ? 

40.  If  A  is  ^  as  old  as  B,  and  if  he  was  eight  times  as 
old  as  B  20  years  ago,  find  the  present  age  of  each. 

41.  A's  age  exceeds  B's  by  21  years.  In  8  years  A  will 
be  1|  times  as  old  as  B.     Find  the  present  age  of  each. 

42.  A's  age  exceeds  B's  by  12  years.  Twelve  years 
ago  A's  age  was  ^  of  B's  age.  Find  the  present  age  of 
each. 

43.  Find  three  numbers  such  that  the  sums  of  the  num- 
bers in  pairs  of  two  are  6,  8,  and  12. 

44.  A  has  $15  more  than  B  ;  B  has  $5  less  than  C; 
A  and  B  and  C  together  have  $65.     How  much  has  each  ? 

45.  A  and  B  and  C  have  $54.  A  has  six  times  as  much 
as  B ;  B  and  C  together  have  as  much  as  A.  How  much 
has  each  ? 

/ 


Ch.  XII,  §  168]  SIMPLE  EQtiATlOKS  Oil 

46.  A  and  B  have  only  |  as  much  money  as  C ;  B  and 
[C  together  have  six  times  as  much  as  A  ;  B  has  $>680  less 
than  A  and  C  together.     How  much  has  each  ? 

47.  A  can  row  6  miles  an  hour  down  a  stream,  and  2 
miles  an  hour  against  the  stream.  Find  A's  rate  in  still 
water,  and  the  rate  of  the  current. 

48.  A  crew  can  row  20  miles  in  2  hours  down  a  stream, 
and  12  miles  in  3  hours  against  the  stream.  Find  the  rate 
of  the  current,  and  the  rate  per  hour  of  the  crew  in  still 
water. 

49.  A  man  can  row  Si  miles  down  a  river  in  56  minutes. 
If  the  river  has  a  current  of  2  miles  per  hour,  find  the  rate 
of  the  man  in  still  water. 

50.  At  what  time  between  3  and  4  will  the  hands  of  a 
clock  be  together?  between  7  and  8?  between  9  and  10? 

51.  At  what  time  between  5  and  6  will  the  hands  of  a 
clock  first  be  at  right  angles  ?  between  6  and  7  ?  between 

10  and  11  ? 

52.  At  what  time  between  12  and  1  will  the  hands  of 
a  clock  be  exactly  opposite  ?  between  4  and  5  ?  between 

11  and  12? 

53.  At  what  time  between  8  and  9  is  the  hour  hand  of 
a  clock  20  minute-spaces  ahead  of  the  minute  hand  ? 

54.  At  what  time  between  4  and  5  is  the  minute  hand 
of  a  clock  exactly  5  minutes  ahead  of  the  hour  hand  ? 

55.  The  sum  of  the  two  digits  of  a  number  is  9,  and 
if  9  be  subtracted  from  the  number  the  digits  will  be 
reversed.     Find  the  number. 


198  ELEMENTARY   ALGEBRA  [Cir.  XII,  §  1(53 

56.  The  tens'  digit  exceeds  the  units'  digit  of  a  number 
of  two  digits  by  1,  and  if  9  be  subtracted  from  the  num- 
ber, the  digits  will  be  reversed.     Find  the  number. 

57.  The  sum  of  the  digits  of  a  number  of  three  digits 
is  17  ;  the  hundreds'  digit  is  twice  the  units'  digit  ;  if  39() 
be  subtracted  from  the  number,  the  order  of  the  digits  will 
be  reversed.     Find  the  number. 

58.  The  sum  of  the  digits  of  a  number  of  three  digits 
is  5 ;  the  hundreds'  digit  is  |  of  the  units'  digit ;  if  the 
number  be  divided  by  the  sum  of  the  digits,  the  quotient 
so  derived  is  8S^  less  than  the  number.    Find  the  number. 

59.  A  number  is  expressed  by  three  digits  whose  sum 
is  18.  If  the  digits  in  the  hundreds'  and  units'  places 
be  interchanged,  the  number  will  be  diminished  by  792. 
The  digit  in  the  tens'  place  is  |-  of  the  sum  of  the  other 
two  digits.      Find  the  number. 

60.  A  can  do  a  piece  of  work  in  3  days,  and  B  can  do 
it  in  5  days.  In  how  many  days  can  A  and  B,  working 
together,  do  the  work  ? 

61.  A  can  do  a  piece  of  work  in  3  days,  B  in  7  days, 
and  C  in  5  daj^s.  How  many  days  will  it  take  all  together 
to  do  the  work  ? 

62.  A  can  dig  a  ditch  in  Ij-  days,  B  in  5^  days,  and  C 
in  Q^  days.  How  many  days  will  it  take  all  together  to 
do  the  work  ? 

63.  A  and  B  together  can  plough  a  field  in  15  days, 
while  A  and  C  together  can  plough  it  in  18  days,  and  C 
in  30  days.  In  how  many  days  can  B  and  C  together 
plough  the  field  ? 


:n.  XII,  §  1G8]  SIMPLE   EQUATIONS  199 

64.  A  and  B  can  build  a  walk  in  6  days,  B  and  C  in 
^i  days,  and  A  and  C  in  10  days.  How  many  days  will 
t  take  A,  B,  and  C  together  to  build  the  walk  ? 

;  65.  A  and  B  can  do  ^  of  a  piece  of  work  in  2  days  ;  B 
'.an  do  I  of  it  in  6  days.  How  long  will  it  take  A  alone 
o  do  J  of  the  work  ? 

66.  Two  pipes,  A  and  B,  can  fill  a  cistern  in  70  minutes, 
^  and  C  in  84  minutes,  and  B  and  C  in  140  minutes, 
low  long  will  it  take  for  each  alone  to  fill  it  ? 

67.  One  tap  will  empty  a  vessel  in  80  minutes,  a  second 
n  200  minutes,  and  a  third  in  5  hours.  How  long  would 
b  take  to  empty  the  vessel  if  all  the  taps  were  open  ? 

68.  A  and  B  can  do  a  piece  of  work  in  m  days,  B  and  C 
a  n  days,  and  C  and  A  in  p  days.  How  many  days  will 
:  take  A,  B,  and  C,  all  working  together,  to  do  the  work  ? 

69.  A  cistern  can  be  filled  by  two  pipes  in  5  and  7  hours 
espectively,  and  can  be  emptied  by  a  third  pipe  in  a  hours. 
n  what  time  can  the  cistern  be  filled  if  the  first  two  are 
unning  into,  and  the  third  is  emptying  the  cistern  ? 

70.  A  train  runs  100  miles  in  the  same  time  that  a 
3Cond  train,  whose  rate  is  3|  miles  an  hour  less,  runs 
0.  miles.     Find  the  rate  of  each  train. 

71.  Two  trains  leave  A  at  the  same  time,  and  run  in 
pposite  directions.  The  first  train  runs  at  a  rate,  in 
liles  per  hour,  j  faster  than  the  second.  How  man)^ 
ours  will  each  train  have  run  when  they  are  425  miles 
part,  if  the  distance  covered  by  the  first  train  in  10  hours 
xceeds  that  covered  by  the  second  train  in  8  hours  by  120 
liles  ? 


200  ELEMENTAUY  ALGEBRA  [Ch.  Xll,  § 

72.  A  and  B  are  240  miles  apart.  If  at  the  same  time 
a  train  leaves  A  and  B,  and  runs  for  the  other  place,  how 
far  from  A  will  they  meet  if  the  train  from  A  runs  at  the 
rate  of  45  miles  an  hour,  and  the  other  ^  as  fast  ? 

73.  A  leaves  the  place  X  at  8  a.m.,  and  2  hours  latei 
B  leaves  Y,  100  miles  from  X,  and  meets  A  at  noon.  Ij 
A  had  left  at  8.30  a.m.,  and  B  at  9  a.m.,  they  would  alsc 
have  met  at  noon.     Find  the  rate  of  A,  and  of  B. 

74.  A  is  100  units  east  from  B.  If  A  and  B  mov( 
toward  each  other,  they  will  meet  in  4  minutes  ;  but  i:. 
both  move  west,  A  overtakes  B  in  20  minutes.  Find  theii: 
rates  of  speed. 

75.  A  left  a  certain  town  and  travels  at  the  rate  o 
a  miles  an  hour,  and  in  n  hours  was  followed  by  B  at  th« 
rate  of  h  miles  an  hour.  In  how  many  hours  did  B  over 
take  A  ? 

76.  A  leaves  New  York  and  travels  at  the  rate  of  1. 
miles  in  5  hours  ;  8  hours  after,  B  leaves  New  York,  anc 
travels  after  A  at  the  rate  of  1*3  miles  in  3  hours.  Ho^^ 
far  must  B  travel  to  overtake  A  ? 

77.  A  and  B  run  a  mile.  First,  A  gives  B  a  start  o 
44  yards  and  beats  him  51  seconds ;  in  the  second  heal 
A  gives  B  an  allowance  of  1  minute  15  seconds,  and  is  beatei 
by  88  yards.     Find  the  time  it  takes  B  to  run  a  mile. 

78.  A  fox  is  pursued  by  a  hound.      The  fox   takes 
leaps  while  the  hound  is  taking  3|^.     Four  of  the  hound'; 
leaps  are  equivalent  to  7  of  the  fox.     The  fox  has  45  c 
her  own  leaps  the  start.      How  many  leaps  will  each  mak 
before  the  fox  is  caught  ? 


Cii.  XII,  §  163]  SIMPLE   EQUATIONS  201 

79.  Find  the  principal  upon  which  the  simple  interest 
for  3  years  and  3  months  at  3|^  is  $93.60. 

80.  Find   the  time  required  for  $2275  to  amount  to 

^2378.74  at  3|/o. 

81.  Find  the  rate  per  cent  at  which  $20,000  doubles 
itself  in  27  years,  9  months,  and  10  days. 

82.  A  sum  of  money  at  simple  interest  in  5  years 
amounted  to  $2400,  and  in  7  years  to  $2560.  Find  the 
principal. 

83.  A  has  twice  as  many  4:Jo  bonds  as  5^  bonds,  whose 
[)ar  values  are  each  $1000.  The  bonds  produce  an  annual 
income  of  $1950.  Find  the  number  of  4^  and  of  5^ 
l>onds. 

84.  A  has  $20,000  invested  between  real  estate  and 
stocks,  the  par  value  of  each  share  being  $100.  On  the 
r(jal  estate  he  nets,  at  5| /o,  $440  ;  on  the  stocks,  at  3|^, 
he  nets  $8  less  than  on  the  real  estate.  Find  the  amount 
ill  stocks. 

85.  The  sum  of  A's  income  for  3  years  at  simple  in- 
terest on  $12,500,  and  on  $15,000  for  4|  years  at  simple 
interest,  is  $4020.  If  the  rates  of  interest  were  inter- 
changed he  would  receive,  in  the  same  time,  $3975.  Find 
the  different  rates. 

86.  The  sum  of  the  capitals  of  A,  B,  and  C  is  $120,000. 
A's  capital  is  invested  at  3|^,  H's  at  4/o,  and  C's  at  3|/o, 
and  the  sum  of  their  incomes  is  $4530.  If  the  rates  at 
which  A's  and  B's  capitals  are  invested  are  interchanged, 
the  income  of  all  is  $30  less.     Find  their  capitals. 


202  KLEMEN  TAIIY   ALGEBRA  [Cii.  XIT,  §  IG:] 

87.  A  mass  of  gold  and  silver  which  weighs  10  pounds 
loses,  when  weighed  in  water,  -^^  of  itself.  If  gold  loses 
^ig,  and  silver  ^^  of  its  weight,  when  weighed  in  water, 
how  many  pounds  of  gold  and   silver  are   there  in   tlie 

^  mass  ? 

88.  A  mass  of  tin  and  copper,  which  weighs  in  air  687 
pounds,  weighs  in  water  608|  pounds.  If  one  pound  of 
tin  loses  ^|§  of  a  pound,  and  one  pound  of  copper  loses 
2^2^2  ^f  '^  pound,  when  weighed  in  water,  how  many  pounds 
of  tin  and  copper  are  there  in  the  mass  ? 

89.  If  a  number  of  soldiers  be  formed  in  a  solid  square, 
24  men  fail  to  get  places  ;  but  if  another  solid  square  be 
formed,  with  one  more  man  on  a  side,  there  are  29  places 
unfilled.     Find  the  number  of  soldiers. 

90.  How  many  ounces  of  14  carat  gold  must  be  mixed 
with  40  ounces  of  15  carat  gold  to  make  a  mixture  of  14  i 
carat  gold  ? 

91.  Five  pounds  of  gold  840  points  pure  are  melted 
with  7  pounds  of  another  sort,  and  produce  a  mass  700 
points  pure.     How  many  points  pure  is  the  second  sort  ? 

92.  How  many  quarts  of  water  must  be  mixed  with  250 
quarts  of  alcohol  80 ^/o  pure  to  make  a  mixture  75^  pure? 

93.  A  piece  of  work  can  be  done  by  20  workmen  in  11 
days,  and  by  30  master  workmen  in  7  days.  In  how  many 
days  can  the  work  be  done  by  22  workmen  and  21  master 
workmen  ? 

94.  At  a  gathering  of  14  men  and  23  women  the  ratio 
of  unmarried  men  to  unmarried  women  is  2  to  5.  Find 
the  number  of  married  couples  present. 


CHAPTER  XIII 

INEQUALITIES 

164.  The  signs  >  and  <  express  inequality :  a>h  is 
read  "a  is  greater  than  S"  ;  a<h  is  read  "a  is  less  than 
&/'  Two  quantities,  a  and  6,  can  be  compared  in  three 
different  ways:  (1)  a  =  h^  (2)  a  >  b^  (3)  a<b.  When 
a>h^  a—h  is  positive;  when  a  <h^  a  —  h  is  negative. 
In  general,  a  quantity  is  said  to  be  greater  than  a  second 
quantity  when  the  first  quantity  less  the  second  quantity 
is  positive  ;  and  a  quantity  is  said  to  be  less  ^han  a  second 
quantity  when  the  first  quantity  minus  the  second  quan- 
tity is  negative.  Since,  by  §  20,  all  positive  quantities  are 
greater  than  zero,  if  a  >  J,  then  a  —  5  >  0 ;  and,  since  all 
negative  quantities  are  less  than  zero,  if  a  <  5,  a  —  J  <  0. 

165.  An  inequality  is  a  statement  that  one  of  two  ex- 
pressions is  not  equal  to  (that  is,  is  greater,  or  less  than) 
the  other.  The  first  member  of  an  inequality  is  the  ex- 
pression to  the  left  of  the  sign  of  inequality  ;  and  the 
second  member  is  the  expression  to  the  right  of  that  sign. 

Thus,  o?  -f-  W  is  the  first,  and  2  ah  the  second,  member  of  the 
inequality,  o?  +'b'^  ^2  ab, 

A  term  of  an  inequality  is  any  term  of  either  the  first 
or  second  member.  Two  inequalities  subsist  in  the  same 
sense  when  they  have  the  same  sign  of  inequality. 

Thus,  a  >  6  and  c>  d  are  inequalities  subsisting  in  the 
same  sense. 

203 


204  ELEMENTARY   ALGEBRA  [Ch.  XIII,  §  106 

Inequalities  subsist  in  the  opposite  sense  when  they  have 
opposite  signs  of  inequality. 

Thus,  a>  b,  c  <  d,  are  inequalities  which  subsist  in  the 
opposite  sense. 

166.  The  general  principles  upon  which  inequalities 
rest  are  : 

I.  If  equals  be  added  to  unequals^  the  sums  are  unequals 
subsisting  in  the  same  sense. 

If  a>b,  (1) 

then  a-b>0.  (2) 

Now,  (a+c^-(b  +  e}  =  a-b,  (3) 

or,  substituting  (3)  in  (2), 

(a+c)-Cb  +  c^>0,  (4) 

or,  rewriting  (4),  a+  c  ^b  +  c.  (5) 

II.  If  equals  be  subtracted  from  unequals^  the  remainders 
are  unequals  subsisting  in  the  same  sense. 

If  a>b,  (1) 

then  •  a-J>0.  (2) 

Now,  (a—  c)  —  (b  —  c)  =  a  —  by  (3) 

or,  substituting  (3)  in  (2), 

(^a-e^-(b-c)>Q,  (4) 

or,  rewriting  (4),  a—  c^h—  c.  (5) 

Application  of  I  and  II  :  Any  quantity  in  an  inequality 
may  be  transposed  from  member  to  member  if  the  sign  of 
that  quantity  be  changed. 

If  a—c^b^  (1) 

by  I,  a>b^c.  C2) 


Cii.  XIII,  §  166]                     INEQUALITIES  205 

If                                           a+h>c,  (1) 

by  II,                                               a>c-b.  (2) 

If  the  sic/7is  of  all  the  terms  of  an  inequality  be  changed^ 
the  sign  of  iiie quality  must  be  reversed. 

If  a  — J>c  — c?,  (1) 
transposing  all  the  terms  in  (1), 

d—e>b  —  a^  (2) 

or,  rewriting  (2),                    b—a<d—c,  (3) 

III.  Ifunequals  be  subtracted  from  equals^  the  remainders 
sid)sist  in  the  opposite  sense. 

If                                                      b>c,  (1) 

rewriting  (1),                           6  —  ^  >  0,  (2) 

changing  all  signs  in  (2),      c  —  J  <  0.  (3) 

Now,  (a-b)  +  (-a+c)==  -b  +  c,  (4) 
substituting  (4)  in  (3), 

(^a-b)  +  (-a  +  c}<0,  (5) 

rewriting  (5),                         a  —  b  <,  a  —  c.  (6) 

IV.  If  unequals  be  multiplied  by  positive  equals^  the 
products  subsist  in  the  same  sense. 

I     If                                                   a>b,  (1) 

then                                           a-b>0.  (2) 

Let  m  be  any  positive  quantity.  Then  m(^a  —  J)  must 
\)o  a  positive  quantity,  since  the  product  of  two  positive 
([iiantities  must  be  positive. 

Tlierefore,                     m(a  —  J)  >  0,  (3) 

or,  rewriting  (3),             ma  —  mb:>  0,  (4) 

or,                                               ma  >  mb.  (5) 


206  ELEMENTARY   ALGEBRA  [Ch.  XIII,  §  166 

Since  the  process  of  division  is  multiplication  by  the 
reciprocal  of  the  divisor,  it  follows  from  IV  that  if 
unequals  be  divided  by  positive  equals  the  quotients 
subsist  in  the  same  sense. 

Application  of  IV  :  To  clear  an  inequality  of  fractions 
multiply  each  term  by  the  L,  C,  D,  taken  as  a  positive 
quantity. 

Thus,  if  _^4_^>^  (1) 

multiplying  (1)  by  24,  —6x  +  Sx>x.  (2) 

V.  If  unequals  be  multiplied  by  negative  equals^  the 
products  subsist  in  the  opposite  sense. 

If  a>b,  (1) 

then  a-b>0.  (2) 

Let  —71  be  any  negative  number.  Then  —n(a—b) 
must  be  a  negative  quantity,  since  the  product  of  a  nega- 
tive and  a  positive  quantity  is  a  negative  quantity. 

Therefore,  -  n  (a  -  J)  <  0,  (3) 

or,  rewriting  (3),  — /^^  +  n5  <  0,  (4) 

or,  —naK—  nb.  (5) 

Since  the  process  of  division  is  multiplication  by  the 
reciprocal  of  the  divisor,  it  follows  from  V  that  if  un- 
equals be  divided  by  negative  equals  the  quotients  sub- 
sist in  the  opposite  sense. 

Henceforth,  in  this  chapter^  literal  quantities  are  used  to 
represent  only  positive  and  unequal  quantities.  Tliis  fact 
must  be  kept  in  mind,  for  otherwise  the  proofs  will  not 
hold. 


jCh.  XlII,  §  167]  INEQUALITIES  207 

167.  A  conditional  inequality  is  true  only  for  some 
value  or  values  of  the  letters  involved.  An  absolute 
inequality  is  true  for  all  values  of  the  letters  involved. 

Thus,  2a;  —  3>a;  +  2  is  a  conditional,  and  a^  +  b^>2  ab  is 
an  absolute,  inequality. 

A.  Prove  that  a^+b'^>2ab. 

Either  (1),  a-b>0,  or  (2),  a-b<0. 

1.  If  a-^>0,  (1) 
multiplying  (1)  by  itself,  a^  -  2  ab  +  b^>0,  (2) 
transposing  in  (2),                          a^  +b^>2  ab,  (3) 

2.  If  a-b<Oy  (1) 

multiplying  (1)  by  itself,  a'-2ab  +  b'>0,  (2) 

(1)  is  negative :  multiplying  a  negative  number  by  itself  is, 
by  V,  an  inequality  subsisting  in  the  opposite  sense. 

Transposing  in  (2),  cr  +  ^-  >  2  ab. 

B.  Prove  that  a^  +  6^  >  ab  (a  +  6). 

Now,                                a^-2a^4-^'>0,  {A) 

transposing  —ab  in  (J[),      a^  —  ab  +  6^  >  ab,  (1) 
multiplying  (1)  hj  a-\-b, 

(a  +  b)  (a?  -ab-\-  Ir)  >ab(a  +  5),  (2) 

a^  +  W->ab{a  +  b).  (3) 

C.  Prove  that  a^ -\- b'^  +  c^>ab  +  be  +  ca. 

Now,  by  A,  a'  +  b'>2  ab,  (1) 

and,  by  A,  52  _,_  ^2  ^  2  be,  (2) 

and,  by  ^,  c^  +  a'>2ca,  (3) 

adding  (1),  (2),  and  (3),     2  (a'  -{-b'+c'):>2  (ab  +  bc  +  ca),  (4) 

dividing  (4)  by  2,  a^ +  b"+ c"  >ab  +  bc  +  ca,  (5) 


208  ELEMENTARY  ALGEBRA  [Ch.  XIII,  §  168 

B.    Prove  that  a^  +  b^+ c^>Z  abc. 

ISTow,  by  (7,  o?  +  h^-\-c^>  ah  +  hG  +  ca,  (T 

multiplying  (1)  by  a,    a^  +  ab^  +  ar  >  crb  +  abc  +  a%  (2 

multiplying  (1)  by  b,    a^b  -\-b^-\-  bc^  >  ab'  +  b^c  +  abc,  (3 

multiplying  (1)  by  c,     ah  +  Wc  +  c^  >  abc  +  6c-  +  c^a,  (4 
adding  (2),  (3),  and  (4),  and  uniting, 

a^  +  W-\-&>^abc.  (6 

The  type  forms,  A^B^  (7,  and  2>,  should  be  remembered* 

168.    The  solutions  of  various  problems  in  conditiona 
inequalities  are  illustrated  in  the  following  problems. 

1.  In   the   conditional   inequality,  3a;  +  |>a;  +  8,  fine 
one  limit  of  x. 

Let  3  a;  + 1  >  a?  +  8.  (1) 

Multiplying  (1)  by  3,  9  a;  +  4  >  3  a;  +  24,  (2)' 

transposing  and  uniting  in  (2),      ^x'>  20,  (3) 

dividing  (3)  by  6,  x>  3i  (4) 

2  X 

2.  In  the  conditional  inequalities,  (1)  :?:  +  7  >  — -  +  9, 

o 

(2)   — -  <  -  +  2,  find  the  integral  values  of  x. 

Multiplying  (1)  by  3,         3  a?  +  21  >  2  a?  +  27,  (3) 

transposing  and  uniting  in  (3),  a:  >  6,  (4) 

multiplying  (2)  by  20,  8  a;  <  5  a^  +  40,  (5) 

transposing  and  uniting  in  (5),      3  a;  <  40,  (6) 

dividing  (6)  by  3,  x<  13\,  (7) 

From  (4)  and  (7),  x  lies  between  the  amits  6  and  13^ ;  and 
may  therefore  take  the  integral  values,  7,  8,  9,  10,  11,  12, 18. 


Ch.  XIII,  §  168]  INEQUALITIES  209 

EXERCISE  LXXXV 

1.  Between  what  limits  must  x  lie,  to  satisfy  the  in- 
equalities  2a;-3>20  and  32;-7<22:  +  6? 

2.  Given  2x—^<x+b,  and  ll  +  2x<'ix+b,  find  the 
limits  of  X. 

3.  Giyen^^  +  ^+3a:<14,  and  ^^~^  +  2a;>9,  find 
the  limits  of  x. 

4.  Given  3a;— 5>2a;+l,  and  3a;+15>4:r+5,  find 
the  limits  of  x. 

Prove    the    following    inequalities,    the    letters    being 
])ositive  and  the  sign  ^^  being  read,  "not  equal  to '^ 

5.  f  +  ^>2,ifa=^5. 
0      a 

6.  a^>2ah-V^,  ii  a--^h. 

7.  77^2  _|_  ffi-\.  p^::>mn  +mp  +  np^  ii  m^n,  n^p^  m^p, 

8.  a%^  +  Pc^  +  (T^a^  >  3  aWc^,  if  a^-h,  a=^c,h^c. 

9.  an  +  hm  <  1,  if  a^  +  J^  =  1  and  if  nfi+rfi^l  and  if 
a^n^  and  h4^m, 

10.  aa;+  %<15,  if  ^2+  52=  25  and  if  a;2+  /=  5. 

11.  2a3  +  63^a(a2  +  a5  +  ?^2)^  if  ^-^5. 

12.  a3-J3>3^j(^_5),  if  ^>6. 

13.  (a +  5)3+  ((?+d:y>(a  +  5  +  c?+(^)(a+J)(c?  +  c?), 
if  (a  +  6):?t(c  +  d). 

14.  ^2  +  4  62  +  ^2  ^  2  aJ  +  2  5^  +  ^c,if  a^2h,a^c,2h-^c. 

m      n      p     n  ^  m  ,  p     .J,      ^  ^  -,       ^ 

15.  _ J |_^->_j f--^    if  m>n^  n>p^  and  m>p^ 

p      m     n     p      n      m 


CHAPTER   XIV 

INVOLUTION   AND   EVOLUTION 
INVOLUTION 

169.  The  operation  of  raising  an  expression  to  any 
given  power  is  called  involution.  An  expression  is  said 
to  be  expanded  when  the  indicated  multiplications  have 
been  performed. 

Thus,  (a)^  and  (a  +  &)-  have  been  expanded  when  the  re- 
spective products  have  been  found  to  be  a^  and  a^  +  2ab  +  W, 

MONOMIAL!^ 

170.  Involution  of  monomials  is  subject  to  the  follow- 
ing Index  Laws,  in  the  proofs  of  which  a  =?^  0,  and  m  and 
n  are  restricted  to  positive  integers. 

I.  {a'^'Y  =  a''''\ 

By  definition,  (a"")"  =  [(a  to  m  factors)  to  n  factors], 

by  associative  law,  =  a  to  mn  factors, 

by  definition,  =  cf''. 

The  exponent  of  the  powei*  of  any  given  monomial  is  found 
hy  multiplying  the  exp>onent  of  the  given  monomial  by  the 
index  of  the  required  power, 

II.  (ahy^=aH'^. 
By  commutative  and  associative  laws, 

(ab)""  =  (a  to  m  factors)  (b  to  m  factors), 

by  definition,  =  a"*6"*. 

210 


(II.  XIV,  §171]       INVOLUTION  AND   EVOLUTION  211 

Similarly,  (abc)"^  =  a'^b"'c'^. 

The  mill  power  of  the  product  of  two  quantities  is  equal  to 
the  product  of  their  mth  powers. 

\bj        6^*    ' 
By  commutative  and  associative  laws, 

j  -  J  =  (a  to  m  factors)  -f-  (b  to  m  factors), 

by  definition,  =  a"*  h-  Z>"*  =  — • 

The  mth  power  of  the  quotient  of  two  quantities  is  the 
quotient  of  their  mth  powers, 

171.    Involution  is  also  subject  to  the  Law  of   Signs. 
(— a)(  — «)  =  (  — ^)2_^2^ 
(a)  (a)  =  (a)2  =  a^, 
(  — a)(  — ^)(  — a)  =  (  — a)^=  —  a^,  etc. 

All  even  powers  of  a  negative  moyiomial  are  positive^  while 
all  odd  powers  of  a  negative  monomial  are  negative ;  all 
powers  of  a  positive  monomial  are  positive. 

EXERCISE  LXXXVI 

Expand  the  following  expressions : 

1.  (a^^.  6.   -(-4^(^)^      ^^     (-11^53)4 

2.  ia^y.  ■7-    (2a;V0^-  '       i^a%y 

3.  (-  a^y.        «•  (-  2  "^"y^'y-   12.  -  r^-^Y- 

9.    _(_  4:^5^)6.  \    2ac   j 


3 


212  ELEMENTARY   ALGEBRA     [Ch.  XIV,  §§  172, 173 

BINOMIALS 

172.  The  expansion  of  binomials  may  be  shortened  by 
employment  of  the  Binomial  Theorem,  a  proof  of  which  is 
given  in  Chapter  XXIV.  The  use  of  this  theorem  is  evi- 
dent from  the  following  type  forms,  which  are  derived  by 
multiplication : 

(a  +  hy  =  d'  +  2ah  +  h^,  (1) 

(a  +  6)3  =  a^  +  3  arh  +  3  aZ^^  +  W,  (2) 

(a  +  hy  =  a^  +  4^  a%  +  6  a^^  +  4  aW  +  h\  (3) 

(a  +  6)^  =  a^  +  5  a%  + 10  a%'  +  10  a%^  +  c>ah^  +  h\  (4) 

(a  +  6)«  =  a«  +  6  a%  +  15  a'ly"  +  20  a^l/  + 15  a^¥  +  6  a6^  +  h\  (5) 

Similarly,  it  may  be  shown  that  the  expansion  of  the 
binomial  (a  —  J)  gives,  if  the  exponents  are  those  of  the 
left  members  respectively,  the  results  in  (1)  to  (5),  except 
that  the  signs  of  the  terms  are  alternately  plus  and  minus, 
the  first  term  being  plus. 

173.  Examination  of  the  expanded  forms  shows,  if  n  be 
the  exponent  indicating  the  pow6r,  and  a  and  b  are  respec- 
tively the  first  and  second  terms  of  the  binomial,  that 

1.  The  number  of  terms  in  the  expansion  is  ti  +  1. 

2.  Every  term,  except  the  last,  in  the  expansion  con- 
tains a.;  and  every  term,  except  the  first,  contains  h. 

3.  The  exponent  of  a  in  the  first  term  is  n^  and  decreases 
by  1  in  each  succeeding  term  ;  the  exponent  of  h  in  the 
second  term  is  1,  and  increases  by  1  in  each  succeeding 
term. 

4.  The  first  coefficient  is  1,  the  second  n ;  the  third,  and 
any  subsequent  coefficient,  is  derived  from  the  preceding 


Cft.  XIV,  §  178]     INVOLUTION   AND   EVOLUTION 


213 


term  by  multiplying  the  coefficient  by  the  exponent  of  a 
and  dividing  this  product  by  the  exponent  of  h  increased 
byl. 

Any  binomial  may  be  expanded  by  this  method  if  in 
the  right  member  a  equals  the  first  term  and  h  equals  the 
second  term. 

1.  Expand  {a^-1iy. 
By  type  form  (3),  §  172, 

(a2  -2  &y  =  (a2)4  _  4  {a^f  (2  6)  +  6  {(j?)\2  Vf  -  4  (a')  (2  by  +  (2  b)* 
=  a«  -  8  a%  +  24  a'b'  -  32  a'b^  +  16,b\ 

In  a  similar  way,  a  polynomial,  in  the  form  of  a  binomial, 
may  be  expanded. 

2.  Expand  Cx-2y+S z^. 
By  type  form  (2),  §  172, 

l(x-2y)+3zJ=(x-2yy+3(x-2yy(3z)+3(x-2y)(3zy+(Szy 
=  a^-6  x^y +12  xy^-Sf +9  afz-36xyz+36y''z 
+21xz^-54.yz'+21z\ 


EXERCISE  LXXXVII 

Expand  the  following  expressions  : 

1.  Qp  +  qY^  8.   (2a+iy, 

2.  (^x+yy.  9.    (x+2yy. 

3.  (1  +  ay,  10.   (x^-y'^y. 

4.  (p+qy.        11.  (1-qy. 
5-   (x-yy,  12.   (2x-3yy. 

6.  (h+iy,  13.    (3:^-22/2)4. 

7.  (x  +  yy,  14.   (3  mn  —  ipy 


15.  (2a;2_5y)5. 

16.  (2a^-8b^y. 

17.  (a  —  S  +  6?)3. 

18.  (^a-b-2ey. 

19.  (22:-y+3^)3. 

20.  (a  —  b—  cy. 

21.  (2/2^2— 3:i:^+?/2/. 


214  ELEMENTARY   ALGEBRA      [Cn.  XIV,  §§  174-170 

EVOLUTION 

174.  The  operation  of  extracting  a  root  of  an  expressio] 
is  called  evolution,  and  is  indicated  by  the  radical  sign,  V 
The  quantities  whose  roots  are  to  be  extracted,  calle< 
radicands,  are  written  after  the  radical  sign.  The  par 
ticular  root  to  be  extracted  is  indicated  by  a  small  number 
called  the  index  of  the  root,  written  above  the  radical  sign 
The  index  2  is  generally  omitted.  If  the  index  of  the 
root  is  an  even  number,  the  root  is  called  an  even  root ; 
if  an  odd  number,  the  root  is  called  an  odd  root. 

Thus,  V4,  -\/81,  V?",  are  even  roots ;  "v/g,  -^^^243,  '""^^?^, 
are  odd  roots. 

175.  If  a  quantity  can  be  expressed  as  the  product  of 
two  equal  factors,  one  of  these  factors  is  called  the  square 
root  of  the  quantity ;  one  of  the  three  equal  factors  of  a 
quantity  is  called  the  cube  root ;  and,  in  general,  one  of 
the  n  equal  factors  is  called  the  /ith  root. 

Since  involution  and  evolution  are  inverse  processes, 

176.  The  one  positive  root  of  a  positive  number  is  called 
its  principal  root ;  the  one  negative  root  of  a  negative  number 
is  called  its  principal  odd  root. 

The  radical  sign  will  be  used  to  indicate  the  principal 
roots  only. 

Thus,  V4  means  the  positive  square  root  of  4;. that  is, 
V4=+2;  similarly,  V25= +5;  -^^:::27=-3;  ^^=^243=^3; 
\j  a''  =  a. 

Note.  Only  expressions  whose  exponents  are  multiples  of  the 
indices  of  the  roots  will  be  discussed  in  this  chapter. 


Oh.  XIV,  §177]     INVOLUTION  AND   EVOLUTION  215 

MONOMIALS 

177.    The  Index  Laws  for  the  evolution  of  monomials 
are  the  inverse  forms  of  tlie  Index  Laws  for  involution. 

I.    -v^'o^  =  d"". 

By  I,  §  170,  (d'y  =  d^\ 

)y  definition,  -v^a^*^  =  d^. 


11.    ^al^h'^e^abc. 

By  II,  §  170,  (obey  =  a^5V% 


3y  definition,  ^ d'lfc'^'  =  ahc. 

From  I  and  II  is  derived  the  Rule  for  the  Root  of  a 
Monomial  in  the  form  of  a  Product:  Divide  the  exponent 
jf  each  factor  hy  the  index  of  the  required  root. 

IIL  4f=t 


By  III,  §170,  (|)"  =  |, 


3y  definition, 


j^lw^  __  a 


From  III  is  derived  the  Rule  for  the  Root  of  a  Monomial 
n  the  form  of  a  Quotient:  Divide  the  exponent  of  each 
factor  in  the  terms  of  the  fraction  hy  the  index  of  the 
'e quired  root. 


1.    Simplify  x/^^^i^ 


343  ai2j9 

^f64xY  ^   AT'^y^  J2?xy^ ___^:xf 
^'343a^%'^     ^Va^'^W^       '    " 


7  a'b'     7  a%^ 


216  ELEMENTAKY   ALGEBRA  [Cu.  XIV.  §  178 

EXERCISE   LXXXVIII 

Simplify  the  following  expressions : 


1.  V^.  9.  Vy  a%*c^  17.  Va\x  -  yf 

2.  V4^.  10.  V64  a%^(fi.  18.  V-^3^f\ 

3.  VmV.  11.  -^21  a%K                        ,y-^ 

4.  v^^SP^.  12.  ^16^*P.  ^®-  \  16^' 


12 


5.  -^8-27.  13.    Via+by.  ^1       6^a%^ 

6.  -^^85^2.  14.   Va^-2ab  +  b^,     ^^'  ^~M3^ 

7.  </f^^  15.   a/(^+1)3.  ,  5/      32x^^y^_ 


SQUARE  ROOT  OF  POLYNOMIALS 

178.  Since,  §  173,  a  polynomial  may  be  squared  as  a 
binomial,  (t  +  2i^^  may  be  taken  as  the  type  form  of  the 
square  of  a  polynomial.  Examination  of  the  way  that  the 
square  root  of  t^  +  u(^2  t -{- u)^  which  is  called  the  square 
root  formula,  is  obtained,  will  disclose  a  method  by  which 
the  square  root  of  any  polynomial  may  be  obtained. 


Since  (t  +  u)^  =  t^ -{'2fu  + u^,  VW+¥tu  +  u^  =  t  +  u, 

(A)  The  first  term  of  the  root  is  the  square  root  of  the 
f.rst  term  of  the  formula. 

(B)  The  second  term  of  the  root  is  obtained  by  dividing 
the  second  term  of  the  formula  by  twice  the  part  of  the  root 
already  found. 

The  formula  may  be  applied  to  any  polynomial,  if  i> 
represents  the  part  of  the  root  already  found  and  if  u 
represents  the  next  term  of  the  root. 


CH.XtV,§178i     INVOLUTION  AND  EVOLUtloN  ^it 

1.    Extract  the  square  root  of  4:x^+  4x1/  +  y^. 

Let                              t^  +  2tu  +  u^  =  4:  x^  +  4.xy  +  y^,  (1) 

by(^),                                                t  =  2x,  (2) 

squaring  (2),                                      f  =  4:X^,  (3) 

subtracting  (3)  from  (1),  u(2  t  +  u)  =  Axy  +  y^f  (4) 

by  (5),                                               u  =  y.  (5) 

Substituting  t  =  2xj  and  u  =  y,  in  (4), 

u{2  t  +  u)  =?/(4  x  +  y)r=4.xy  +  y\  (6) 
Since 


V4  aj2  -f.  4  a;?/  +  ?/2  =  V^^  -}-  2  ^^^  +  ^^^  ^  ^  +  u,  (7) 


and  since  ^  =  2  ic,  and  u  =  y,  V4 x^ -]- 4:xy  -\- y' =  2x  +  y:      (8) 
The  work  may  be  more  compactly  written : 


t  =  2x 

4.x'  +  4.xy-hy'\2x  +  y 

4.x' 

2t  =  4.x 
u  =  y 

2t-\-it  =  4:X-{-y 
u(2t  +  2i)  =  y(4.x-{-y) 

4.xy  +  y^ 
4  xy  +  y^ 

The  terms  of  the  polynomial  should  be  arranged  either 
in  ascending  or  in  descending  order  of  some  one  of  its 
letters ;  otherwise  the  formula  method  is  not  available. 

If  the  polynomial  contains  more  than  three  terms,  it 
should  be  carefully  noticed  that  the  part  of  the  root 
already  found  in  every  case  is  represented  by  t. 

Since  (a  +  h -cy=\_a^(h  -  c)]^  =  [(a  +  ?>)  -  cf, 
and  since       {t  +  iif  =  (a  +  6  —  cf, 
t  is  represented  successively  by  a  and  a-^b. 


218 


ELKMENTAUY   ALGEBUA  [Cii.  XIV,  §  178 


2.    Extract  the  square  root  of  a^  + i  c^  +  P—2ab +  4:  be 
—  4:  ae. 


t  =  a 

\a  —  b  —  2c 
a^-2ab-4:ac  +  b'-  +  4:bc-\-4  c' 
a' 

2t  =  2a 
u  =  ^b 
2t-Yu  =  2a-b 
u(2t  +  u)==-b(2a-b) 

~2ab-4  ac  +  b^  +  4:bc  +  A  r 
~2ah            +b^ 

2t  =  2a-2b 
u  =  -2c 
2t  +  u  =  2a-2b-2c 
u(2t+u)=-2c(2a-2b-2c) 

—  4:ac         +  4  &c  +  4  c- 

—  4  ac         +  4  6c  +  4  c- 

In  the  above  example,  after  the  second  term  of  the  root  has 
been  found,  the  first  two  terms  are  together  equal  to  t..  Since 
t=(a—b),  andi  has  been  squared  and  subtracted,  the  remainde': 
again  corresponds  to  the  expression  u(2 1  +  w). 


EXBBOISB    liXXXIX 

Extract  the  square  roots  of  the  following  expressions : 

1.  25a2-70ac  +  49c2. 

2.  a^  +  2ab  +  b^+2ac  +  2bc  +  (^. 

3.  b^  +  2hc  +  e^-2ab-2ac  +  a^. 

4.  4:  a^ +  12  ab +  9  P +  16  ac +  2i  be +16  c^. 

5.  49a^  +  ib^-28ab  +  42ac  +  9<fl-12bo. 

6.  9  a^  +  80  a%  +  19  a%^  + 40  ab^  + 16  b*. 

7.  89  a%^  -10ab^  +  16a*-  56  a%  +  25  h*. 

8.  4  a,«  -  1 2  a*b  -  28  aW  +  9  a^^  +  42  ab*  +  49  5«. 

9.  49  m^  +  4  w2  +  16  jo2  +  28  mn  +  16  np  +  56  mp. 
10.  a*b^c^  +  a%*c^  -  2  a%^c^  -  2  a%\^  +  aWc^  +  2  aW<?. 


II.  XIV,  §§170, 180]     INVOLUTION  AND  EVOLUTION 


219 


179.  The  extraction  of  the  square  root  of  an  expression 
Mjntaining  fractions  is  often  made  easier  by  arranging  the 
;erms  in  descending  order  of  some  one  of  its  letters. 

1      ^  2       1 

Extract  the  square  root  of  a^  +  -~-  —  -  +  2 1 — . 


t^a 


2^  =  2a 

1 

a 

a 

i(2t+u)  =  ^(2a  +  -\ 
a\          a) 

2 

+ 

1 

2t  =  2a  +  - 

_2 

-V- 

a 
1 

a 

a^     a* 

2t  +  u  =  2a  +  ^--\ 

.(2<+.)  =  -i(2a  +  ?- 

i-) 

_2 
a 

_2  +  l 

a    cv    a^     a* 
a? 


a     w 


180.  Under  certain  conditions  the  formula  method  may 
3e  applied  to  polynomials  not  in  the  form  of  perfect 
quares.  These  conditions  are  discussed  in  the  following 
ixample.  The  square  roots  of  such  expressions  are  called 
ipproximate  square  roots.  If  the  polynomial  has  a  true 
square  root,  the  square  of  the  root  equals  the  given  poly- 
aomial ;  if  the  polynomial  has  an  approximate  square  root 
mly,  the  square  of  the  root  plus  the  remainder  equals  the 
^iven  polynomial. 


220 


ELEMENTARY  ALGEBRA  [Ch.  XIV,  §  ISd 


The  symbol  •••is  called  the  symbol  of  cojitinuation^  and 
is  read,  '^and  so  on." 

ThuSj  x^x^  -f-  a^  •  •  •  is  read,  '^  x  +  x^  +  :(?  and  so  on.'^ 
1.    Extract  the  approximate  square  root  of  1  +x. 


1  + 
1 

■X 

^2      8 

t=l 

2t  =  2 

X 

"=1 

2i+„=2+| 

«(2i  +  .)  =  |(2  +  |) 

< 

2(  =  2  +  a; 

a? 

«  =  -- 

4 

2t  +  u  =  2+x-t 

u{2t+u)  =  -%(2+x- 
8  V 

-f) 

x"     a^      a;* 
4      8      64 

y/T+. 


-V[C 


o;^     a;2 


2^\  ,  a?      x^~\ 


4      4       8   '  64/  '    8      64j' 


1  +  :^  +  ^-:^-:^  +  ^    +T7-^ 


The  square  root  is  extracted  by  the  formula. 

In  the  above  example  the  root  is  not  approximate  if  tlie 

remainder,— —/^^  larger  algebraically  and  mimerically  than 

1+x.  In  general,  approximate  square  roots  are  to  be  inter- 
preted  as  approximate  square  roots  of  such  expressions  only 
as  produce  remainders  less  than  the  given  expressions. 


Ch.  XIV,  §180]     INVOLUTION  AND   EVOLUTION  221 

EXERCISE  XO 

Extract  the  square  roots  of  the  following  expressions : 

b'  a^ 

_      «»       4a^     2452  ,  36J« 


a     a^     a**     a* 


V^       c       <?•       h        a       0? 

9  g*     8  52     4  56     6  a3     13  a2     8  5* 
■     66        3        9a2       6*  ■•■    62    +3^- 


W'  .        iiJ    l/C/ly        .        \j  rj   VVIIV  rj    lyllV       . 


V^       hd       cP      2bn      2  dn      16  ^i^ 

7.  ^4  +  4^  +  10^2  +  13^  +  13  +  1+  25  ^11^ 

X       4:X^       x^       x^ 

8.  4  +  121^  +  ^I^+n^  +  ^^^  +  ^^+lO.:. 

64  4  16  16  4 

9.  a4  +  2a3  +  a2  +  2a  +  4  +  ?  +  l  +  ^^  +  l. 

a     a^     a^     a^ 

10.  x8  +  22:6  +  :i:^  +  2:r2  +  4  +  ^  +  ^  +  4  +  -^. 

rr^      a;*      rr^      o:^ 

11.  :^^_2x3+32:2-4a;+5--  +  4-^  +  4• 

•    X      ar      ^2:;^      a;* 

Express  to  four  terms  the  apjjroximate  square  roots  of 
tfie  following  expressions : 

12.  1  --  X.  14.    1  —  a:  +  x^.  16.    x^  +  X, 

13.  a2  +  6.  15.   x^-\-x^  +  \.  17.   x^  +  Sx+2. 


222  ELEMENTARY   ALGEBRA     [Ch.  XIV,  §§  181,  182 

ARITHMETICAL   SQUARE    ROOTS 

181.  Square  roots  of  arithmetical  numbers  may  be  ex- 
tracted by  the  formuhi  method. 

Since  VI  =  1,  VIOO  =  10,  VIoTKW  =  100,  Vl,OUO,000  = 
1000,  etc.,  the  square  root  of  a  number  >1  and  <100 
has  one  digit,  the  square  root  of  a  number  >100  and 
<  10,000  has  two  digits,  the  square  root  of  a  number 
>  10,000  and  <  1,000,000  has  three  digits  ;  and  so  on.  If, 
therefore,  tlie  number  be  separated  into  periods  of  two 
digits  each,  running  from  right  to  left,  the  number  of 
periods  will  equal  the  number  of  digits  in  the  root. 


Thus  ViT64  has  two  digits,  V811,801  has  three  digits. 

182.  Every  integral  number  may  be  considered  as  made 
up  of  tens  and  units.  Hence  (t+  u}^^  where  t  represents 
the  part  of  the  root  already  found  and  u  represents  the 
next  term  of  the  root,  will  correspond  to  any  integral 
number  in  the  form  of  a  perfect  square. 

42  =  40  +  2  =  ^  +  ^^,  (1) 

squaring  (1),         (42)^  =  (40  -j-  2)^  ={t  +  u)',  (2) 

simplifying  (2),   1764  =  1600  +  160 +  4:  =  f +  2tu  +  u^  (3) 
indicating  square  roots  in  (3), 


V1764  =  V1600  +  160  +  4  =  -y/t'  +  2fAi  +  u\  (4) 

1.    Extract  the  square  of  1764  =  1600  +  160  +  4. 


^  =  40 
f  =  1600 


2^  =  80 

u=    2 

2t  +  u==:S0-\-2 

i((2t  +  u)  =  2(S0  +  2) 


1600  +  160  +  4  I  40  +  2  =  42 
1600 


160  +  4 
160  +  4 


Ch.X1V,§182J     involution   AND    EVOLUTION 


223 


The  work  necessary  in  writing  a  number  in  the  form 
f  -\-2tu+  (r  is  tedious,  and  may  be  abridged;  the  preceding 
written  in  the  abridged  form  is: 


<  =  40 

17  64  1  40  +  2  =  42 

f-  =  1600 

16  00 

2t  =  80 

164 

u=   2 

2t  +  u  =  S2 

u(2t  +  tt)=2(S2) 

164 

In  the  above  example,  if  ^  =  value  of  the  digit  in  the  tens' 
place,  and  u  =  value  of  the  digit  in  the  units'  place,  t  is  the 
greatest  multiple  of  10  whose  square  is  <  1764 ;  that  is, 
^  =  40.  Subtracting  ^-  =  1600,  the  remainder  is  1G4.  Dividing 
X64  by  2  if  =  80,  the  quotient  is  2,  which  is  u.  Hence  u(2t-{-  u) 
=  2(80  4-  2)  =  164  is  to  be  subtracted  from  the  remainder,  164. 
The  remainder  being  0,  the  square  root  is  40  +  2  =  42.  In  the 
above  example  the  work  may  be  further  abridged  by  omitting 
the  two  zeros  in  the  square  of  40. 


2.    Extract  the  square  root  of  4,414,201. 


t  =  2 

4  41  42  01 

<2  =  4 

4 

2<  =  40 

41 

u=    1 

2  <  +  t<  =  41 

u{2t+u)  =  \{Al) 

41 

2  <  =  420 

42 

u=     0 

2 «  +  w  =  420 

m(2«  + It)  =  0(420) 

0 

2 1  =  4200 

42  01 

u=       1 

2t  +  u  =  4201 

?«(2<  +  m)  =  1C-^201) 



42  01 

2101 


224 


ELEMENTARY   ALGEBRA     [Ch.  XIV,  §§  183, 184 


183.  Since  VO.Ol  =  0.1,  VO.OOUi  =  0.01,  VO. 000001  = 
0.001,  etc.,  the  square  root  of  a  decimal  in  the  form 
of  a  perfect  square  has  half  as  many  decimal  places 
as  the  number  itself.  A  decimal  is  therefore  separated 
into  periods  of  two  digits  each,  running  from  left  to 
right. 

After  pointing  off  the  decimal,  the  square  root  is 
extracted  as  if  the  decimal  were  an  integer. 

1.    Extract  the  square  root  of  0.01301881. 


<=1 

0.01  30  18  81  1  0.1141 

f=l 

1 

2t  =  20 

30 

u=   1 

2t  +  u  =  21 

u(2t  +  u)=l(21) 

21 

2  <  =  220 

918 

M=   4 

2  <  +  M  =  224 

m(2<  +  m)  =  4(224) 

8  96 

2  t  =  2280 

22  81 

u=       1 

2t  +  u==  2281 

M  (2  <  +  «)  =  !  (2281) 

22  81 

Since  there  are  eight  decimal  places  in  the  number  there 
are  four  decimal  places  in  the  root. 


184.  The  approximate  square  roots  of  numbers,  whether 
integral  or  decimal,  or  both,  not  in  the  form  of  perfect 
squares,  may  be  found  by  annexing  zeros  to  fill  out  the 
periods  of  two  digits  each  until  the  number  of  periods 
equals  the  number  of  root  digits  required. 


Ch.  XIV,  §184]     INVOLUTION   AND   EVOLUTION 

1.    Extract  the  square  root  of  7.1  to  three  decimals, 


225 


t  =  2 

7.10  00  00 

t'  =  i 

4 

2<  =  40 

310 

ti=   6 

2  t  +11=4:6 

u(2t  +  u)  =  6(i6) 

2  76 

2i  =  480 

34  00 

u=     6 

2  <  f  M  =  486 

m(2<  +  m)  =  6(486) 

2916 

2  <  =  4920 

4  84  00 

M=         9 

2t  +  u  =  4929 

m(2<  +  k)  =  9(4929) 

4  43  61 

EXERCISE  XOI 

Extract  the  square  root  of  the  following  numbers : 

1.  361.  6.   136,161.  11.   0.1369. 

2.  1681.  7.   3,404,025.  12.   0.134689. 

3.  T396.  8.   1,225,449.  13.   0.094864. 

4.  71,824.  9.   3,466,383,376.    14.  8476.0436. 

5.  15,129.  10.   0.0081.  15.   2499.700009. 

Extract  the  approximate  square  root  to  four  decimals 
)f  tlie  following  numbers : 

1.6.   2.  19.   6.  22.   0.831. 

.7.   3.  20.   7.  23.   10.4. 

.8.   5.  21.   10.  24.   32.701. 


CHAPTER   XV 

RADICALS 

185.  The  quantity  Va  has  already  been  defined,  §  17t 
as  the  quantity  whose  nth  power  is  a,  or  (V^)'*  =  ^.  I 
a  is  an  exact  n\h  power,  the  existence  of  such  a  quantity  i 
at  once  evident,  as  V8  =  2.  But  if  a  is  not  an  exact  ni 
power,  it  becomes  necessary  to  prove  the  existence  of  -yja 
Such  a  proof  is  beyond  the  province  of  this  book  ;  an^ 
a  simple  numerical  example  must  suffice.  It  is  not  pos 
sible  to  obtain  exactly  the  value  of  V2,  since  there  is  n 
number,  integral  or  fractional,  whose  square  is  exactly  2 

^^^'  (1.4)2  <2<(1.5)2,  (^ 

(1.41)2<  2  < (1.42)2,  (^ 

(1.414)2<  2  <  (1.415)2.  {G 

In  (J.),  since    2   lies   between   (1.4)2  and    (1.6)2,  V 

differs  from  1.4  and  1.5  by  less  than  they  differ  from  eac' 

other  :  that  is,  since  1.4  and  1.5  differ  from  each  othe 

by  0.1,  V2  differs  from  either  by  less  than  0.1 ;  similar! 

in  (^),  V2  differs  from  1.41  and  1.42  by  less  than  0.01 

and  in  ((7),  V2  differs  from  1.414  and  1.415  by  less  thai 

0.001.     Continuing  the  process  shown  in  (^),  (^),  an( 

(6^),  a   number   may  be    found  which  will   represent  a 

close  an  approximation  of  V2  as  is  required. 

p 

b  l5  c  D  S         X 

Fig.  11. 
226 


(ii.XV,  §18G]  RADICALS  227 

The  value  of  V2  may  be  represented  graphically.  On  the 
lino  OX,  Fig.  11,  let  equal  distances  be  laid  off  from  0  toward 
the  right,  and  OA  represent  the  number  1,  OB  the  number  2, 
etc.  Then  1.4  will  be  represented  by  OC,  1.5  by  OD.  The 
numbers  1.4,  1.41,  1.414  will  be  seen  to  be  represented  by 
lines  whose  terminal  points  move  toward  the  right,  while  the 
numbers  1.5,  1.42,  1.415  will  be  represented  by  lines  whose 
terminal  points  move  toward  the  left.  The  terminal  points 
representing  these  two  sets  of  numbers  will  approach  each 
other,  but  no  terminal  point  in  either  set  can  cross  into  the 
region  of  the  other.  Yet  the  numbers  show  that  the  terminal 
points  may  be  made  as  near  to  each  other  as  may  be  required. 
There  will  be  some  point  P  which  will  be  the  limiting  posi- 
tion of  both  sets  of  terminal  points;  and  the  line  OP  will 
represent  V2. 

186.    An  indicated  root  of  a  quantity  is  called  a  radical. 

Thus,  Va,  V27,  are  radicals. 

An  expression  which  is  composed  of  radicals  is  called 
a  radical  expression. 

Thus,  -Vx  +  V27,  Va  —  V^,  are  radical  expressions. 

All  integers  and  fractions  are  called  rational  quantities. 

All  other  numbers  are  called  irrational  quantities.  The 
simplest  class  of  irrational  quantities  consists  of  indicated 
roots  which  cannot  be  extracted. 

Thus,  2,  and  |,  are  rational ;  V2,  Vl  +  V2,  are  irrational. 

An  expression  which  contains  rational  quantities  only  is 
called  a  rational  expression. 

Thus,  a  +  f  is  a  rational  expression. 

An  expression  which  contains  an  irrational  quantity  is 
called  an  irrational  expression. 

Thus,  a  +  V2  is  an  irrational  expression. 


228  ELEMENTARY   ALGEBRA       [Ch.  XV,  §§  187-189 

187.    A  radical  whose  radicand  is  rational  and  whose 
root  is  irrational  is  called  a  surd. 


Thus,  -Va  and  -v^4  are  surds ;  while  v  1  -f  V3,  being  the  in- 
dicated root  of  a  quantity  not  rational,  is  not  a  surd. 

The  order  of  a  surd  depends  upon  the  index  of  the  root. 
A  quadratic  surd,  or  a  surd  of  the  second  order,  has  2 
for  the  index  of  the  root ;  a  cubic  surd,  or  a  surd  of  the 
third  order,  has  3  for  the  index  of  the  root  ;  a  biquadratic 
surd,  or  a  surd  of  the  fourth  order,  has  4  for  the  index  of 
the  root,  etc. 

Thus,  Va,  V^,  Vc,  are  respectively  quadratic,  cubic,  and 
biquadratic  surds. 

188.  A  rational  factor  of  a  surd  is  called  the  coefficient 

of  the  surd. 

Thus,  f  is  the  coefficient  of  |  Va5. 

Surds  which  have  1  as  a  coefficient,  expressed  or  im- 
plied, are  called  entire  surds. 

Thus,  -Vay  and  Vi  are  entire  surds. 

Surds  which  have  other  coefficients  than  1  are  called 
mixed  surds. 

Thus,  2VS  and  SVa  —  b  are  mixed  surds. 

A  surd  is  called  a  monomial  surd  if  it  consists  of  a  sin- 
gle surd. 

Thus,  a/^  and  5V3  are  monomial  surds. 

The  sum  of  a  rational,  and  a  surd  quantity,  or  the  sum 
of  two  monomial  surds,  is  called  a  binomial  surd. 

189.  The  difference  between  algebraic  and  arithmetical 
irrational  quantities  should  be  noticed.     Such  a  quantity 


.  XV,  §  100]  RADICALS  229 

1,8  V2  is  an  arithmetical  irrational  quantity  ;  similarly, 
quantities  such  as  Va  are  considered  algebraic  irrational 
[uantities,  although  if  a  =  4,  Va  is  an  arithmetical  rational 
[uantity. 

In  this,  as  in  the  preceding  chapter,  the  principal  roots 
mly  are  discussed,  and  the  quantity  under  the  radical 
lign  is  restricted  to  positive  values. 

Thus,  VJ  =^±2,  but  Vi  =  2.  This  fact  must  be  kept  in 
nind,  for  otherwise  some  of  the  proofs  of  the  principles  will 
lot  hold. 

PRINCIPLES   OF   RADICALS 

190.  I.  The  product  of  the  nth  roots  of  any  number  of 
quantities  is  equal  to  the  nth  root  of  their  products. 

By  II,  §  170,         C^a  Vb  V~cy  =  abc, 

oj  definition,  "Va  VJ  ^c  =  -Vabc. 

If  the  radicand  contains  a  factor  whose  exponent  is  a 
multiple  of  the  index  of  the  root,  the  surd  may  be  simpli- 
fied by  I.     Since  -^'a""  =  a^ 

by  I,  V^  =  V^-^  =  a^. 

1.  Simplify  Vl6. 

^/16  = -v/2^  = -v/2^ -v/^  =  2-v/2. 

2.  Simplify  </W^. 

■V^2¥y  =  ■y/2^  =  -\/2^'  -^2^  =  2  x-\/2^. 

3.  Simplify  V25  a -25  b 

V25a-256  =  -\/25  (a  -  h)  =  VsV^^^  =  5  Va"=r&. 


230  ELEMENTARY  ALGEBllA  [Cn.  XV,  §  11)1 


1. 

Simpl 

V8. 
Vl8. 
V75. 
V27. 

17. 
18. 
19. 

EXE 

ify  the  following 

5.  V48. 

6.  V150. 

7.  -^16. 

8.  ^54. 

5RCISE  ■ 

surds : 
9. 
10. 
11. 
12. 
22. 
23. 
24. 

XCII 

^-  54.      13. 
a/80.           14. 
■v/96.          15. 
^192.         16. 

Va%. 

2. 

V25  ax\ 

3. 

■Vfy  mhi^ 

4. 

a/4  2^^3. 

</2l 
■\/f. 

xy^. 

2^/27^-54  6 

• 

20.  Vl6a;— 16^.  25.    a— 3V4a:2— 8^. 

21.  -^xXa—  b).  26.   (a—  2)Va2— a*. 

191.  The  coefficient  of  a  radical  may  be  introduce! 
under  the  radical  by  raising  the  coefficient  to  the  powe 
indicated  by  the  index  of  the  root.     Since  a  =  -v^a", 

Reduce  the  mixed  surds  2  a V2  6,  and  —  3  axVa^x^  to 
entire  surds. 


2  aV2b  =  V(2  ay  V2^  =  VS  a^^. 


-  3  aaj  Va^a;  =  V  (-  3  aic)^  ^a'x  =  V  -  27  aV. 

EXERCISE   XCIII 

Reduce  the  following  mixed  surdsto  entire  surds: 

1.  2V3.  5.    2-J/5.  9.    -ab^hi, 

2.  3^/2.  6.    3-^2.  10.    a-h^^a^^. 


3.  4-V/2.  7.    -5V4.  11.    (a-J)Va-J. 

4.  2\/4.  8.    —  2a-v/x.  12.    —(^a  —  x)^^x--y. 


H.  XV,  §  192]  RADICALS  231 

192.  When  the  radicand  is  in  the  fractional  form  it  may 
e  made  integral,  by  I,  §  190.     Since  \(-)  =-' 

Note.  (1)"  =  1,  hence  ^/V'  =  1.  It  is  usual  to  omit  both  the 
cponent.of  tlie  power,  and  the  index  of  the  root,  of  1. 

When  the  denominator  of  a  radicand  in  the  fractional 
)rm  does  not  contain  a  quantity  whose  exponent  is  a 
Lultiple  of  the  index  of  the  root,  the  denominator  may  be 
lit  into  such  a  form  that  the  indicated  root  may  be  ex- 
acted by  multiplying  both  numerator  and  denominator 
Y  the  same  quantity. 


Thus, 


vi=vf(i)=Vf=v|vs=i^. 


1.    Simplify -y|- 


A|=Vl)  =  V¥  =  Vive-=|v. 


2.    Simplify  ^'If. 


'9ab 


i\2xy _  sj2xy  _  sl2xy  fia^¥\ 
"Vg  ab  ""  ^'  S^aft  ~  V  3-'a6  \i  a?by 


=4^^^'''''^y=h^^^^' 


232  ELEMENTARY  ALGEBHA  [Ch.  XV,  §  1 

EXERCISE  XCIV 

Simplify  the  following  surds  : 


X 


9.    a/— 

X 


10. 


11. 


I2e 


1. 

2. 

15. 

>'l25c2 

3. 

vi. 

16. 

€• 

4o 
5. 

17. 

^ll- 

6. 

^. 

18. 

<fe- 

7. 

^1- 

19.    \     -^  ^    .  30.     „ 


20 


3^^  21, 


22 


23. 


3/5  d- 

14.  Vifi- 


2 

16"  "■     ^MSafif 


*l    7m     . 

•    ^162a^t/2 

.    ^1     3a3     , 

■    ^Z  2048x^2/ 

s/  21  aJe 

■   '^^m^fiy 

j    a^c     . 

■    '^242x^3 

^/   6a4J5 

2g_    j/32a5^,7 


243 


mx 


27      4  10a6^ 
^8l2;"3/2 


28.  J8(^-^) 
^27  (a:- 2/ 

29.  {[JEM 


«-V:n^- 


32. 

j£±^. 

^x-y 

33. 

n 

34. 

m- 

35. 

s<q. 

36. 

20^iVo- 

37. 

45  ^'S- 

38. 

10a#^^ 

:^H.XV,§193]  RADICALS  23^ 

193.    II.     The  quotient  of  the  nth  roots  of  two  quantities  is 
iqual  to  the  nth  root  of  their  quotient^  or  -;^=  \~i' 

By  III,  §170,         (|D"=2, 

-\/a      »» \a 


jy  definition,  _, , 

1.    Express  VT2 -f-  V5  as  a  single  radical. 


2.    Express  V2  a  -?-  V3  J  as  a  single  radical. 


^-^=VI?=V 


2a  ^     2af3b\ 
3  6      ^.S^UfiJ' 


=  \/'5LV2.3a6  =  J-V6^6, 


36 


EXERCISE  XCV 


In  the  following  radicals,  simplify  the   quotients   ex- 
pressed as  single  radicals : 


1.  VG  -!-  V5.  6.  y/z^i/  -i-  Vlt)  m*n. 

2.  ^4  -H  ^.  7.  a/49  a^y  ^  ^UTi^. 

3.  ~^a^</b.  8.  A/(a-6)2H--v/2^. 

4.  ■v'x-^V^.  9.  V(m  —  w)^-5-  V8  w» 


5.    VS  ab -^  V"  xy.  10.  V32a;«/-^V27a6^ 


234  ELEMENTARY    ALGEBRA       [Cii.  XV,  §§  194, 195 

194.  III.      The  nth  poiver  of  the  Tilth  root  of  any  quayitit'^ 
is  equal  to  the  mth  root  of  the  nth  power ^  or  (j\/ay  =  Va^. 

By  definition,  (Va)''  =  Va  to  ri  factors, 

by  I,  §  190,  =  Vo^. 

Simplify  {VT^y. 
{■V2¥'cy  =  -y/'i2^'  =  -v/2VV  =  -y/Wa}^?  -y/^T^c  =  2 ah  -V^irc 

EXERCISE   XCVI 

Simplify  the  following  radicals : 

1.  (V2)3.  4.  (V2^)3.  7.  <iV-i\x^y>\ 

2.  (a/3)2.  5.    (</3^)3.  8.    iViy. 

3.  (a/2)3.  6.    (v'2~^2J^)2.  9.    (^^:M)2. 

10.  (-v/_4aS2)3.  12.   (c^x^-\-2xy  +  y'^y, 

11.  (-a/^=T^)3.  13.    (^^^^^^"^1^)2. 

195.  IV.     The  mth  root  of  the  nth  root  of  any  quantity  is 
equal  to  the  mnth  root  of  the  quantity^  or  "V -\/^  =  "Va. 

Let  x='^'Va,  (1) 

raising  both  members  of  (1)  to  mt\\  power, 

:c-=Va,  (2)^ 

raising  both  members  of  (2)  to  nth  power, 

2;^'^  =  a,  (3) 

extracting  mnth  roots  in  (3), 

a;  =  "Va,  (4) 

from  (4),  VV^  =  "V^.  (5) 


Cii.  XV,  §  195]  RADICALS  235 

A  surd  is  said  to  be  in  its  simplest  form  when  neither  of 
the  reductions  explained  under  I,  §§  190,  192,  and  IV, 
§  195,  may  be  applied;  that  is,  when  the  radicand  is  inte 
gral  and  contains  no  factor  whose  exponent  is  a  multiple 
of  the  index  of  the  root,  and  when  the  index  of  the  root  is 
as  small  as  possible.  Similar  surds  are  those  which,  when 
reduced  to  their  simplest  form,  differ  in  their  coefficients 
only.  Surds  which  are  not  similar  are  called  dissimilar 
surds. 

Thus,  iVa  and  5Va  are  similar  surds,  and  ^Va  and  h\a 
are  dissimilar  surds. 


Reduce  V27  and  V 243  a^  to  their  simplest  forms. 


-v/27  =  -^3^  =  Vs^'33  =  V3. 


^/243  a'  =  a/3^  a'  =  ^</3'  a'  =  -\/3  a. 

EXERCISE   XCVII 

Reduce  the  following  radicals  to  their  simplest  forms; 


1. 

■^4. 

9. 

'</a^. 

17. 

■^v^. 

2. 

•-^25. 

10. 

'V-a\ 

18. 

■^v^. 

3. 

'<m. 

11. 

a/- 1000. 

19. 

^^a^. 

4. 

^64. 

12. 

^-32. 

20. 

■^</ab. 

5. 

^100. 

13. 

^64. 

21. 

■</K/a^^ 

6. 

^243. 

14. 

^a^K 

22. 

V^^u 

7. 

15. 
16. 

^^. 

23. 
24. 

^^^9 

8. 

V^4a2. 

"v^aVa. 

25 


.   "v^2^2«.  26.    -^x^y^/a^. 


236  ELEMENTAKY   ALGEBRA  [Cir.  XV,  §  lOO 

196.    V.    The  nth  root  of  any  quantity  is  equal  to  the  mnth 
root  of  the  mth  power  of  the  quantity^  or  V a=  'Vo™. 


Since  -y^=V(V^)- 


by  III,  §194,  =VV^, 

by  IV,  §  195,  ^V^. 

By  V,  surds  of  different  orders  may  be  reduced  to 
equivalent  surds  of  the  same  order  —  that  order  being  the 
least  common  multiple  of  the  original  orders. 

Reduce  V2  and  V5  to  equivalent  surds  of  the  same  order. 

Since  ■v/25>-v/8,  ^/5>V2. 

EXERCISE   XCVIII 

Reduce  the  following  surds  to  equivalent  surds  of  the 
same  order : 

6.  -v^T^,  ■\/Jab\ 

7.  4 V2  T^y^  ■\/2  xy\ 

8.  W^,   -sjWf. 


1. 

V^,  -V7\ 

2. 

Va,  Va. 

3. 

V2:r,  ^\x\ 

4. 

^a%,  </aP. 

5. 

V3a2,  '<y2xy. 

9.  </&¥¥,  -VSa^bK 


10.   Vll  axi/,  ■\/4:d^xi/'^. 

Arrange  the  following  surds  in  order  of  magnitude : 

11.  -y/i,  </S.  14.   Vl2,    ^48,  ^''ST. 

12.  V3,  ^2,  ^5.  15.   V|,  -v^,  -v^l. 

13.  ^,  ^10,  '<mo.        16.  v|,  -^,  -v/f. 


Ch.  XV,  §  197]  RADICALS  237 

ADDITION   AND   SUBTRACTION   OF   RADICALS 

197.  Similar  radicals  may  be  added  or  subtracted  by 
writing  the  algebraic  sum  of  their  coefficients  as  the 
coefficient  of  the  common  radical,  taken  as  the  unit  of 
addition.  Each  radical  should  be  reduced  to  its  simplest 
form  before  the  process  of  addition  or  subtraction  is 
begun.  Addition  or  subtraction  of  dissimilar  radicals 
may  only  be  indicated  by  connecting  them  with  the  proper 
signs. 

1.  Simplify  Va^  —  2  a Vo^  +  3 Va. 

Va^  =  Va^  Va  =  a  Va, 

—  2  a^cf  =  —  2  a^a^  Va  =  —  2  a^^a, 

3Va  =  3Va, 

Va^  -  2  a^b  +  3  Va  =  (a  -  2  a^  +  3)  Va. 

2.  Simplify  2  VJ  -  4  Vl  +  8  V2. 

8V2=  %-V2, 

2  Vi  -  4  Vi  +  8  V2  =  8  V2. 

EXERCISE   XCIX 

Simplify  the  following  radical  expressions : 

1,  2V^-4VS  +  Vx.  3.    V32  +  V8-Vi28. 

2.  3Va-4Va+2Va.  4.    VT6  +  V54  -  VT28. 

5.    2V^+5V^-a;VT^. 


238  ELEMENTARY   ALGEBRA  [Ch.  XV,  §  197 

6.  4  a^'M  -  VWM  -  a^243  a%. 

7.  -y/xy  +  \  ^xy  +  ^y/xy, 

8.  a'\/x  —  ^J~€^-\-la^a^x. 


9. 
10. 
11. 
12. 
13. 


V16^  -  V25^  +  V49"u. 

V16^  +  VSla  +  V144  a%^. 

'Va%c  +  '\'  aW'c  +  -yjabc^. 
-\/a%c  +  ^ah^c  +  ^  ab(?. 

14.  Vax^  +  VJ^;^  +  V^:z^. 

15.  Vo^  +  '2-yJ~cfWc  -  b-y/ab^. 

16.  V9^^  +  V25^e^  +  VM^^sp^. 

17.  V9^-^'l6^+3V25^- ViOO^. 

18.  Va^  +  3VI(ra^-V64"a^-2V9^. 

19.  1_V3  +  4V5+2V9  +  V108-V80. 
L    Vl8^  +  Vl47^-2V32^-Vl92^  +  V72^. 
.    3  V:?;(a  +  bf  -  VTS  -  VWWi  -  V(a  +  6)2^:. 

V4:i:— 4?/+ Vl6x— 16  2/+5V49:r-49«/ 

J  V^  _  (a  +  J)  V^3  +  a V^  +  S  Vo^. 

__.  -^54-^128+^250.       28.    3^1-2^81  +  6^3. 

26.  VI+fV2-V|.  29.    2av^-^3-8A/^^ 

27.  2V3-VS  +  fVJ.  30.    VX_|V8  +  4V|. 
31.  J^zi_^  +  v^2z:;a2  +  V(^2Tr^. 

^a;+  a 


20 
21 
22 

23. 
24. 
25. 


DH.XV,§ie8]  RADICALS  239 

MULTIPLICATION   OF   RADICALS 

198.    The  product  of  any  two  monomial  radicals  may  be 
"ound  by  applying  V,  §  196,  and  I,  §  190. 

1.  Multiply  2V5  by  SVi. 

2V5  =  2-s/W  =  2-\/l2E, 
3^  =  3a/2"*  =  3v'16. 
2V5  .  3^4=6\/2000. 

2.  Multiply  S-Va^  by  4</b^. 


S-Vai^  =  3  VCo^"  =  3  VaV, 


4\/6r' =  4  V(&ar*)'  =  4V6V, 
3  v^2  •  4a/6^  =  12V^mc^=  12^/^  ^/^m?  =12  ic'-v/o^ft^. 


EXERCISE  C 

Find  the  products  of  the  following  radicals ; 

1.  Va  .  </6.  10.   </a^  ■  '</al 

2.  V^.-v/^.  11.  V2.-\/8. 

3.  ^6  .  ^.  12.    ^12  .  ^2.  20.  #  .  #. 

21.  v'72  .  •v'TOS. 

22.  </5^  •  </2j^. 

7.  </a  •  -^b.  16.  </¥^  .  '-H^.     23.  ^51^  •  ^10^. 

8.  </^  .  ^y.  17.    ^  .  ^|.  24.  ^1024  .  </^. 

9.  v'^  .  ^^z.         18.  </-Y^  •  ^/J^.         25.  \/1024^3  .  ^^2. 


4.  ^5.^/2.  13.   ^a;.^a;8. 

5.  Va  •  Vs.  14.   ■^x'  •  ^y. 

6.  Va  •  Vx.  15.   V2^  •  -s/Jx. 


240  ELEMENTARY  ALGEBRA  [Ch.  XV,  §  19! 

MULTIPLICATION  OE    POLYNOMIALS    INVOLVING  RADICAL! 

199.  The  product  of  two  or  more  polynomials  involving 
radicals  is  found  in  the  same  way  that  the  product  of  t\v( 
rational  expressions  is  found  ;  the  terms  of  the  polynomiali 
being  expressed  in  their  simplest  form  before  the  procesi 
of  multiplication  is  attempted. 

Multiply  V60  +  VM  by  V27  -  V8. 

V60  +  V24  =  2  Vl6  +  2  V6, 
V27-V8   =:3V3   -2V2, 

2V15  +  2V6 
3V3-2V2 


6V45  +  6  Vl8  -  4  V30  -  4Vl2  = 
18  V5  + 18  V2  -  4  V30  -  8  V3. 


EXERCISE  CI 

Multiply  the  following  expressions  involving  radicals: 

1.  2  ay/x  —  Q^a^xy  by  2 Va^  +  5  ay/y. 

2.  4V8-V32  +  2V6()by  V2. 

3.  V75  -  V150  +  2  V243  by  V3. 

4.  V|+V|  +  V|by  V6  +  V6  +  V8. 

5.  2Vf7-3V75  +  4Vl2  by  2V3  +  5V48  +  4V5. 
3 V12  +  V32  +  2 V80  by  4 V45  -  2V48  -  2 Vl8. 
3V27  -  5V20  -  2V343  by  2V48  -  3  V45  -  3VlT2 
§  VT8  +  4  V.54  -  4  V75  by  4  V48  +  2  Vl62  -  3  Vl50 
2^16  +  4-^24  -  3^108  by  2^^54  -  3^/81+  4^32. 

10.    4\/24  +  3\/256-5\/l35  by  3v'81-^/32  +  2^625 


6. 
7. 
8. 
9. 


Cii.XV,§200]  RADICALS  241 


DIVISION   OF   RADICALS 


200.    The  quotient  of  any  two  monomial  radicals  may 
l)c  found  by  applying  V,  §  196,  and  II,  §  193. 

Divide  </T2  by  V2  ;  and  8V2^  by  2-</^. 


2</^        2</^       2V  a^      2\x<\xj     2\x'\  2x\       ' 


EXERCISE   Oil 

Find  the  quotient  of  the  following  radicals : 

1.  Va^-\/b.  4.   ■Va-i--\/a.  7.  ■\/x^-i--^jfi. 

2.  a/^^v/^.  5.   A/a -5-^.  8.   V30-5-\/45. 

3.  Va  -H  V5.  6.   Va^  -^  Va*.  9.  a Va  -i-  ■\I <fi. 

11.  3V5H-A/i5.  18.  2\/686^5V56. 

12.  12V27^v'9.  19.  10  V216  H- 2^/36. 

13.  4^49^3V7.  20.  a/I024^2v'192. 

14.  4^100  ^VlO.  21    <S^I2-VS^S 

15.  3^/90  H- 2^/18.  , .         , 

16.  3V343H-A^i9.  ■  ^(a-by  '  ^Ca-b/ 


242  ELEMENTARY   ALGEBRA  [Ch.  XV,  §  201 

DIVISION   OF    POLYNOMIALS   INVOLVING   RADICALS 

201.  The  quotient  of  a  polynomial  involving  radicals 
by  a  monomial  radical  may  be  found  in  the  same  way  as 
in  rational  expressions. 

The  division  of  two  polynomials  involving  radicals  is 
best  effected  by  a  process  called  division  by  means  of 
rationalization ;  by  this  process  the  radical  denominator  is 
transformed  into  a  rational  quantity.  The  least  factor, 
the  product  of  which  and  the  given  radical  produces  a 
rational  quantity,  is  called  the  rationalizing  factor. 

Thus,  the  rationalizing  factors  of  V2,  -y/'a^,  V<x'^  are  respec- 
tively V^,  -y/a'X-,  Va. 

Two  binomial  quadratic  surds  in  the  forms  Va  +  V^ 
and  -\/a  —  VJ  are  called  conjugates  of  each  other.  Since 
(Vo;  +  V2/)( V.r  —  Vt/)  =  X  —  y^  the  rationalizing  factor  of 
a  quadratic  binomial  surd  is  its  conjugate.  It  is  necessary 
to  multiply  both  the  terms  of  the  fraction  expressing  the 
quotient,  by  the  rationalizing  factor, 

2 

1.  Rationalize 

V3 

V3      V3     V3     ^ 

2.  Rationalize  — _  '    '  _- 

■\/a-\--Vb  _  Va  +  V^  .  Va  +  V^  ^  a4-2Vq£+& 


H.  XV,  §  201]  RADICALS  243 

EXERCISE   cm 

liationalize  the  denominators  of  the  following  fractions  : 
5  a  __  1 

1.    — :r'  6.    — -'  !!•    -' 

V5  Aa  3  +  V2 


12  _  -.o         1 

3V7 


7V3 
3.  -^^.  6V5  13. 


4. 


16. 


17. 


18. 


20. 


5V3 

7. 

48 

5V32 

8. 

a  +  h 

d 

Va  +  6 

Z7. 

^a-b 

10. 

10  +  V21 

V7  +  V3 

V5  +  V3 

V5-V3 

V8-V6 

V8  +  V6 

a2_aV6  +  6 

a-y/b 

a  —  b 

•v'a+2J-V3J 

^y 

_. 

12. 


5V3 


^  14. 


^4  V6 

•\/b+G 
a 


bVab'  ^b  +  Vc 

4Vl8  ^g     a  +  V6 

5V12  '   a-V^ 

22.    _A±i_. 

^a  +  ^6 

■Va  +  Vb 
a  +  Vab  +  b 

1Q  +  4V5 

I  +  V2  +  V5' 

4 

I  +  V2  +  V3 

12 

V2-V3  +  V6" 

14V5 


23. 


24. 


19.    t^--ttv»i-t^  25. 


26. 


21.    —  ^ —  ■  27. 


244  ELEMENTARY   ALGEBRA  [Ch.  XV,  §  202 

SQUARE   ROOT   OF  A  BINOMIAL   QUADRATIC   SURD 

202.  The  square  root  of  a  binomial  quadratic  surd,  in 
the  form  of  a  perfect  square,  may  often  be  extracted  by 
inspection  if  it  be  remembered  that  the  binomial  quad- 
ratic surd  is  a  disguised  form  of  oP^+lab^lP',  Since 
(Va  + V6)2=  6^+ 2Va6  4- J  =  (^+ &)+ 2Va6,  the  square 
root  of  (a+  5)+  2^\fab  is  found  by  obtaining  two  quan- 
tities whose  sum  i^  a+h  and  whose  product  is  ah.  Since 
(V2  +  V3)2  =  2  +  2V6  +  3  =  5+  2V6,  the  square  root  oi 
6  +  2 V6  is  V2  +  Vs. 

The  binomial  quadratic  surd  may  be  still  further  disguised 
by  the  introduction  of  the  coefficient  of  ^ab  under  the  radical 
thus  :  (a  +  Z>)  +  V4  ah.  Before  finding  the  square  root  it  is 
necessary  that  the  term  corresponding  to  2^ ah  shall  be  writter 
with  the  coefficient  2. 


1.    Extract  the  square  root  of  8  +  V 60. 


Vs+Veo  =  Vs  +  2  Vl5  =  V3 + V5. 
2.    Extract  the  square  root  of  V24  +  V25. 


V  V24  +  -yj'lh  =  V2vfT5  =  V3  +  V2. 


EXERCISE   CIV 


Extract  the  square  root  of  the  following  binomial  quad- 
ratic surds : 


1. 

3+2V2. 

6. 

9-2VI4. 

11. 

V121-V120. 

2. 

4  +  2  V3. 

7. 

11  -  2  V2i. 

12. 

V64-V28. 

3. 

7  +  2  VlO. 

8. 

11-2V28. 

13. 

V256-V156. 

4. 

8  -  2  Vl5. 

9. 

V121-2V10. 

14. 

J^— JVI4. 

5. 

6  +  2  V5. 

10. 

V8T-V80. 

15. 

2 -Vs. 

H.  XV,  §203]  RADICALS  245 

RADICAL  EQUATIONS 

203.    An  equation  which  involves  the  indicated  root  of 
le  unknown  is  called  a  radical  or  irrational  equation. 


Thus,  V3  -\-x  =  2  is  a  radical  equatiouo 

A  radical  equation  which  involves  square  roots  only  can 
ften  be  solved  as  a  simple  equation  by  isolating  one  or 
lore  of  the  radicals  and  rationalizing  the  resulting  equa- 
on  by  squaring.  But  since  two  equations  with  different 
gns  may  give  the  same  result  when  squared,  the  solution 
btained  by  solving  the  squared  equation  does  not  neces- 
irily  satisfy  the  given  equation.  It  is  necessary  to  test  the 
olution  in  every  case  hy  substituting  in  the  given  equation. 

If  the  equation  contains  a  single  radical,  it  is  simpler  to 
jolate  the  radical  and  then  square  the  resulting  equation; 
t  the  equation  contains  two  or  more  radicals,  the  more 
ivolved  radical  is  isolated.  The  squared  equation  should 
[len  be  simplified,  especial  care  being  taken  to  reduce 
ae  resulting  equivalent  equation  to  the  simplest  integral 
Drm. 

1.    Solve  the  equation :  Va;+  6  +  ■\/x—  2  =  4.  (1) 


Transposing  in  (1),     Vaj  +  6  =  4  —  -yjx  —  2,  (2) 

^uaring  (2),  x  +  6  =  16- 8Vaj-2  +  x-2,        (3) 

L'ansposing  and  uniting  in  (3), 


8V^^=^  =  8,  (4) 

ividing  (4)  by  8,  V^^=^  =  1,  (5) 

quaring  (5),  a;  — 2  =  1,  (^) 

ransposing  and  uniting  in  (6),  x  =  S,  (7) 

Verification  :  V9  +  Vl  =  4 ;  therefore  3  is  a  root  of  (1). 


246  ELEMENTARY  ALGEBRA  [Ch.  XV,  §  203 ; 


2.    Solve  the  equation  :   ■Vx+  6  —  -Vx  —2  =  4.  (1) 


Transposing  in  (1),     Vif  +  6  =  4  +  Vx  —  2,  (2) 

squaring  (2),                        x  +  6  =  16  +  8 VoT^  +  x-2,  (3) 

transposing  and  uniting  in  (3), 

8V^^=^  =  -8,  (4) 

dividing  (4)  by  8,            Vx^^  =  - 1,  (5) 

squaring  (5),                         a?  — 2  =  1,  (6) 

transposing  and  uniting  in  (6)jX==3,  (7) 

Substituting  in  (1),  V9  —  Vl  ^  4 ;  therefore  3  is  oiot  a  root 
of  (1). 


3.    Solve  the  equation  :   ('a  —  b}\——-  +  b  =  a.  (1) 


a  —  b 


Transposing  in  (1),  (a  —  h) \/ =  a—b,  (2) 


dividing  (2)  by  a  -  6,  yj^  =  1,  (3) 

squaring  (3),  — ^  =  1,  (4) 

multiplying  (4)  by  a  —  5,  x  =  a—b,  (5) 

Verification :  (a—b) \h^-^ — |-6  =  a;  a  —  b  +  b  =  a. 
^a  —  b 


4.    Solve  the  equation :   V2  +  a;+ Va;  — 3=  V4a;— 3.    (1) 


Squaring  (1),   2  +  a^+2Va.•--.^'-6+.^-3  =  4a;-3,  (2) 

transposing  and  uniting  in  (2),  2-\/x^—x—6=2  x—2,  (3) 

dividing  (3)  by  2,  V?  -  ic  -  6  =  a;  - 1,  (4) 

squaring  (4),  x'  —  x  —  Q>  =  x-  —  2x-^  1,  (5) 

transposing  and  vmiting  in  (5),  x  =  7,  (6) 

Verification  :  VO  +  V4  =  V25. 


Cii.  XV,  §  203]  KADICALS  247 

EXERCISE  CV 

Solve  the  following  radical  equations : 

1.  -Vx+5  =  S.  -  4.  V7:r+2  =  4. 


2.  6Va:  +  4  =  ll.  5.  ■V5-{-x=3-^/x. 

3.  7  =  3V^-4.  6.  Vl5  +  a;=3V5-V^, 


7.  V22:+ll  +  V2:^-5  =  8. 

8.  V27:r+1  =  2-3V3^. 


9.   V4  +  a;  V24  +  2;2  =  :?;  +  2. 


10.  V^_  +  V^^5. 


11.  Vx  +  Va  +  a; : 

12.  V^  +  V3  +  a; :        . 

-y/S  +  x 

13.  Va;  +  4  a6  =  2  a  +  V^. 


'Va  +  X 
24 


14.  V:r+  ^  =  a  —  -\/x  —  a. 

15.  h  —  a-\/x  =  ^/€fix. 


16.  —  = —  ' 

^x+'i      ■\/x+Q 

17.  x=  a  —  ^ a?"  —  x-y/x'^  +  8  a?. 

18.  V5  +  22:=  V2(8  +  92:)-Vl  +  8a;. 


19.  3Vl  +  2:i:- V8:t:-15=  V2(a;  +  6). 


20.    V9a:-14  +  3V^+2  =  2V92:-2. 


248  ELEMENTARY  ALGEBRA  [Ch.  XV,  §  203 


REVIEW   EXERCISE 

CVI 

Simplify  the  following  expressions 

: 

1.  ^f 

16. 

^|. 

29. 

(^/ax'y. 

'■4- 

17. 

€■ 

30. 

-^</^. 

-^t 

18. 

31. 

^49b 

3/  a 

*•  Vs.- 

19. 

20. 
21. 
22. 
23. 
24. 

25. 

26» 
27. 
28. 

32. 

33. 

34. 
35. 

36. 
37. 
38. 
39. 
40. 
41. 
42. 

^  25a 

5.  (-v/5)3. 

6.  \/40a;3. 

7.  </4Sa^- 

8    ■v/</27. 

\a  +  x_ 
^x  —  a 

(  V2  a)\ 

^</2x^. 

V3.V2. 

9.   \/49a;2. 

X 

^V36a:5. 

10.  </Mx^. 

11.  ^^^18. 

V2-Vi 
V3 

0  , — — -              4  /~r-rr 

12.  V3.^/9. 

13,  </J^  .  V^. 

V5-V2 
4^20  H- Vs. 
%  .  V9x. 
a/J+-v/168. 

-v/21-^\/12. 

14.    A/2--V14. 

(V-a  +  6)*. 

15.   -^j- 3  a/49. 

V21-4Vi7 

43.    Solve  for  a 

5V^  +  4__3 
5Vi-4     2' 

44.   Vi8  +  12V2  +  (2-V3)(2  +  V3). 


CHAPTER  XVI 

IMAGINARIES 

204.  An  indicated  even  root  of  a  negative  number  is 
called  a  pure  imaginary  quantity.  Quantities  which  are 
not  imaginary  are  called  real.  Since  (+2)2  =  (—2)^=  4, 
V—  4  9^  ±  2,  or  the  even  root  of  a  negative  number  is  defined 
as  impossible;  that  is,  V— 4  can  be  expressed  neither  as 
a  positive  nor  as  a  negative  quantity.  Therefore,  pure 
imaginary  quantities  must  be  excluded  from  the  number 
system,  which  up  to  this  point  includes  rational  and 
irrational  quantities ;  or  the  number  system  must  be  en- 
larged to  include  pure  imaginary  quantities.  Because 
such  quantities  are  frequently  met  in  the  solutions  of 
quadratic  equations.  Chapter  XVIII,  and  elsewhere,  they 
are  included  within  the  number  system. 

205.  It  will  be  assumed  that  the  fundamental  laws,  the 
commutative,  associative,  etc.,  govern  operations  of  expres- 
sions involving  imaginary  quantities.  The  proof  of  these 
laws,  as  applied  to  such  expressions,  is  beyond  the  province 
of  this  book. 

206.  The  general  form  of  a  pure  imaginary  is  V  —  a. 
Since  ^ab  =  'VaVb^  similarly,  V— a  =  VaV— 1,  the  lat 
ter  being  called  the  typical  form  of  a  pure  imaginary. 

The  sum  or  difference  of  a  real  and  a  pure  imaginary 
quantity  is  called  a  complex  quantity. 

Thus,  a  +  '\r—  ^  is  a  complex  quantity. 

249 


250  KLEMENTARY  ALGEBRA     [Cii.  XVI,  §§  207,  208 

In  all  operations  ivith  complex  quantities  the  pure  imagi- 
nary should  first  be  reduced  to  the  typical  form. 

Thus,  2V^=1  =  2V4V^^  =  4V^^. 

207.  By  §  175,  (V^=^)2  =  -  J  ;  or  (V^ri)2=  -  i. 
The  form  V  —  1  is   generally   written   in    the    shorter 

form  /. 

The  following  list  gives  some  of  the  integral  powers 
of  i: 

i  =  /,  i^  =  i^  •  i  =  i, 

^3  =:i^  .  i==  ^ {^  f  =  ^6    i=  —  2, 

i4  =  ^2  .  ^2  ^  _f.  ;l^  ^8  ^  ^4  .  ^4  =  +  1^  ^tC.  J 

i 
That  is,  in  general,  z"  =  z^+^.  | 

208.  Imaginary  and  complex  quantities  can  be  employed 
in  the  various  operations. 


1.  Reduce  V— 1(J,  V—  x^^  V—  4:x^y  to  the  typical  form. 


2.  Add  V^^16,  V-  25,  and  V-36. 


■>/=^=V25^  =  5^, 
VZTie  +  V"^^^^  +  V^=^  =  (4  +  5  -j-  6)/  =  15  i. 


(11.  XVI,  §  208]  IMAGINARIES  251 

3.   Multiply  V^^  +  V2  by  2  V^4  +  3 V2. 

2i+V2 
4  i  +3V2 

+  GV2i  +  6 


8  »2  +  10V2  i  +  6 
:-8  +  10V2  i  +  6  =  10V2^•  -  2. 


4.  Divide  1  by  V^^  +  V-  3. 

1  V2 1  -  V3 1  ^  V2  ^•  -  V3  ^•  ^  V2  I  -  V3 1 

V2i+V3*'V2j-V3i~    2i^-3t^  -2  +  3    ' 

=(V2-V3)i. 

5.  Expand  ( V^^  +  V^^y. 
( V3  i  +  V5  if  =  [?■( V3  +  V5)]2  =  *2(a/3  +  V5)2 

=  _1(3  +  2Vl5  -f-  5)  =  -  8  -  2Vi§. 

6.  Extract  the  square  root  of  1  +  2V—  6. 

Vl  +  2V6i=V2«-  +  V'3. 

EXERCISE   OVII 

Reduce  the  following  pure  imaginary  quantities  to  the 
;ypicai  forms : 


1.  V^^25.  4.    yP:^.  7.    V-225/. 

2.  V^;^.  5.    V-  x!^y\  8.    V-484a:8. 


3.    V-100.  6.    -\^--i:3^y\  9.    V-625a^«/2. 

Simplify  the  following  expressions : 


10    V^;n^  +  V-25  +  V-a4-5v'-ioo. 


11.    V^;^ly  +  V-121-V-169-V-196. 


252 


ELEMENTARY  ALGEBRA  [Ch.  XVI,  §  20$ 


12.  V^a;2-V-4a;2- V-9  a:^  +  V- 25x2. 

13.  3+V^^  +  5V^n^  +  16  +  7V^"225. 


14.    a  +  by/-x^+2a-  bV-  x^  +  Sa-i  6V-  x^ 


15. 


^^^  •  V-  9. 


18.  V-i-V-i. 


16.  V^:^25  •  V^^36. 

17.  V^)  •  V^^16. 

21.  (V^a,  +  - 

22.  (  V^^  + 


19.     —  V- 


v/-  rr* 


20.    V—  a^^^  •  V—  ab*. 
^)(2V:^+3V^). 


23.  V—  a^J  •  V—  ^252  .  y_  ^53, 

24.  V^2  •  V^^  •  V^^. 

25.  (i+v^r4)2. 

26.  (  V^^  +  V^)2. 

27.  (2  V^^  +  3  V^=^)2.  32.     (1+V^^)5. 


30.  (V=^+l)8. 

31.  (1  +  V^=T6)*. 


28.  (3V-9-4V"^)2. 

29.  (1-V^^)3. 


33.    (V-|  +  V-i)a 


^7 

V^^Ti 

V-9 

1 

38. 

V-16 

Q 

39. 

V-9 

4.0 

9 

34.     (V-J^ 

35.  (V^:4+V^^+V-16)2. 

36.  ( V  -  a^  -  V^p  _  ^Zr^yi, 

Va 


i)^ 


41. 


42. 


-\/  —  7? 


43. 


V-49 


V^:^ 


44. 


-V~25 
1 


45. 


46. 


47. 


48. 


2  +  V- 

-9 

a  +  V- 

^h 

a  — V- 

~-b 

a-V- 

rft 

a+ V- 

^6 

1 

V2+V-2 


Ch.  XVI,  §  208] 

54.VZ3 

3+2V3-5' 

8  -  5V^^ 
6  +  SV^s' 


IM  AGIN  A  HIES 


253 


49. 


50. 


51. 


52.    —= 


V2+V3  +  V-5 


53. 


54. 


ViTi 


V3  +  V7+V-10 

l  +  V2  +  V^:^* 


55.  (2V3-V^)(4V3-2V^^). 

56.  (a;-5  +  2V^)(a;-5-2V^^). 

57.  (V^--2  +  V^3)(V^  +  2-V^^). 

58.  (a;-2V5  +  3V^=^)(a;-2V5-3V^^). 

59.  (2  -  V^  - 3 V^^)(4 V^^  +  6 V'^2). 

60.  {x-\^  1 V^) {x-\-  1  V^3). 

61.  (a;-2-V3)(a;-2  +  V3)(a;-3  +  V^^) 

(a;-3-V^=n[). 

62.  (a;-l-A/^2)(x-l+V^^)(a;-2  +  V"^) 

(a;-2-V^=^). 

63.  Vi  +  v^^ .  Vi_v^T  •  V3_v^2  •  V3+V^2; 
5.3.4 


64. 


:  + 


:  + 


4-V-4 '  1+V-l  '  1-V^l 

65.  (V^T)2+(v^^)3+(v^^)4+(V^=i:)5 


+(V-i)«+(V-iy+(V-i)8. 


CHAPTER  XVII 

THEORY  OF  EXPONENTS 
THE  EXPONENT  IN  THE   FORM    OF   A   POSITIVE    FRACTIO^^ 
209.    In  §  177  it  was  shown,  if  m  and  n  are  integers  and 

n. 

71  is  a  multiple  of  m,  that  Va"  =^  a"^.  If,  however,  n  is  not 
an  exact  multiple  of  m,  there  can  be  no  meaning  attached 

n 

to  a"*  according  to  the  previous  definition,  §  7,  of  an  ex- 
ponent. Thus,  it  is  impossible  to  speak  of  a^  as  meaning 
a  taken  three-fourths  of  a  time  as  a  factor.  The  definition 
of  an  exponent  is  therefore  extended  to  include  the  expo- 

nent  — ,  it  being  understood  that  a'^  (where  n  and  m  are 
m 

positive  integers  and  a  is  a  positive  real  quantity)  is  siinply 
an  alternative  way  of  writing  V^%  or  the  principal  value 
of  the  wth  root  of  the  nth  power  of  a. 

This  extension  of  the  definition  of  an  exponent  is  valid 
only  in  case  exponents  in  the  form  of  a  positive  fraction 
conform  to  the  laws  of  exponents  which  have  been  shown 
to  hold  for  positive  integers.  That  is,  exponents  in  the 
form  of  a  positive  fraction  must  be  shown  to  obey  the  laws, 

a'^ay  =  a^+^,  I 

a'^ -^  a'^  =  a'^-y,  II 

(a-^)?/  =  a^y,  III 

(ahy^^^a'^h''.  -  IV 

254 


Cii.  XVII,  §  210] 

THEORY   OF   EXPONENTS 

210.   I. 

n     r            n    r 

By  definition, 

«     r 

a'"  a'  =  Va"  Va'", 

by  V,  §  196, 
by  I,  §  190, 

=  V««'v'a'»'-, 

=  Va«»+""', 

ns+mr 

by  definition, 

=  a    ^*    , 

n    r 

or, 

=  a^'*  *. 

II. 

n           r            w    r 

By  definition, 

n            r             

by  V,  §  196, 
by  II,  §  193, 

=  -<Ja-^^'^<Ja^-r^ 

^-^^ns-mr^ 

ns-mr 

by  definition, 

=  a    ^    , 

or, 

III. 

n    r           nr 

By  definition, 

n             

w    r                        r 

jy  conditions, 
by  definition, 
by  III,  §  194, 

(a»'y  =  (A/a")', 

=  V(V<)'-, 

=  VVa''% 

by  IV,  §  195, 

r^V^-, 

by  definition, 

255 


266  ELEMENTARY   ALGEBRA         [Ch.  XVII,  §  211 


IV. 

(a6)'«  =  a'"h'". 

By  definition, 

(ahy  =  V{ah-)\ 

by  III,  §  194, 

=  Va"6% 

by  I,  §  190, 

=  V^V6^, 

by  definition, 

n    n 

It  may  also  be  shown  that  a^  =  a*^. 

By  definition, 

n 

by  V,  §  196, 

=  'Va*", 

by  definition. 

A   NEGATIVE   INTEGER   AS   EXPONENT 

211.  If  m  and  :^  are  integers  and  a  is  a  positive  real 
quantity,  the  quotient  of  aJ^  -^  aJ"  is  aJ^~'',  \i  m> n,  there  is 
no  difficulty :  but  if  m<n,  a  quotient  with  a  negative 
integer  as  exponent  is  obtained ;  as,  —  p,  where  p=zn  —  m, 
or  —  p  =  m—  n.  Such  a  quantity  as  a~^  has  no  meaning 
according  to  the  original  definition  of  an  exponent. 
But  it  is  convenient  to  extend  still  further  that  definition, 
and  to  speak  of  —  j?  as  an  exponent. 

Ji  p  =  n  —  m^  a^  -T-  a^^  =  —  =  a^~^  =  a~^ ;   if  the  numeral 
tor  and  denominator  of  the  fraction  —  be  divided  by  a^\ 
the  quotient  is  evidently =  —     Hence  a~'^  =  —  : 

a  quantity  with  a  negative  integral  exponent  is  equal  to  the  rd 
ciprocal  of  the  same   quantity  with  an  equal  positive   exponen 


€n.  XVII,  §§212,213]     THEORY   OF   EXPONENT^  257 

212.  It  is  necessary  to  sliow  that  negative  integral  ex- 
ponents conform  to  the  laws  of  exponents  for  positive 
integers. 

I.  a-Pa-'i  =  a-^-^. 

^    II.  a-P-T-a-^^a-P-^^. 

a^      a^     a^ 


III.  {a^P)-9  =  a 


pq 


Qa-py    fry    j_ 


(a-p)-«  =  --L-  =  -— —  =  -_  =  aP9. 


IV.    (abyp=a-Ph-P. 


THE  EXPONENT  IN  THE  FORM  OF  A  NEGATIVE 
FRACTION 

213.    If  m  is  a  positive  integer  and  7i  is  a  negative  inte- 
ger which  is  not  an  exact  multiple  of  m,  a  being  a  real  quan- 

tity,  a  '^  may  be  defined  as  the  alternative  form  of  Va~^« 
Exponents  in  the  form  of  a  negative  fraction  must  be 
shown  to  conform  to  the  index  laws  for  exponents  in 
the  form  of  positive  fractions.  Such  exponents  are  made 
positive  by  application  of  §  211 ;  and  hence,  by  §  210, 
obey  the  index  laws. 


258  ELEMENTARY   ALGEBRA     [Ch.  XVII,  §§  214, 215 

214.    It  now  remains  to  prove  in  general  that 


a"  =  a" 


A  negative  integral  exponent  and  a  negative  fractional 
exponent  can  be  made  positive,  respectively,  by  §§  211 
and  213.  It  is  therefore  necessary  to  prove  V  only  for 
positive  fractional  exponents. 


a"  =  a" 


By  definition, 


by  III,  =a^^'^  \ 

The  index  laws  hold  when  one  exponent  is  integral  and 
the  other  is  fractional,  since  the  integer  may  be  written  as 
a  fraction  whose  denominator  is  1. 

ZERO   AS   EXPONENT 

215.  The  case  of  a  zero  exponent  naturally  arises  when 
any  quantity  is  divided  by  itself: 

—  =  ^n-n  ^  ^0 .  but  —  =  1.     Hence  a^  =  1. 

It  is  seen  that  the  value  which  must  he  attached  to  any 
finite  quantity  with  the  zero  exponent  is  unity. 

Zero  as  an  exponent  may  be  shown  to  conform  to  the 
ordinary  laws  of  exponents. 

Note.  It  has  been  shown  in  the  preceding  articles  that  any  rational 
quantity  can  be  used  as  an  exponent.  Examples  in  which  irrational 
numbers  are  used  as  exponents  are  given  in  Chapter  XXV. 


Cii.  XVII,  §  215]         THEORY   OF   EXPONENTS  259 

EXAMPLES 
1.  a^  r=  Vo^. 

8-9-12+18 

3.  25^  .  8-*  =  (52)i,  (2^)-*  =  5.2-2  =  5.4=54  =  ?- 

2  4     4 


3-3„-|j,6\-i      /3-3a-t6V*     3ai6-2     3  •  2^1^     12  a^ 


•  V  64  ;    (,  2«  ; 


EXERCISE  CVIII 

Change  each  of  the  following  expressions  into  radicals : 


1.  ai 

1 

5.  a». 

9.  3a^  +  ji 

2.  ai. 

6.   a-^. 

10.  5b-i-2cK 

3.  a/>-3. 

7.  86-i 

11.   (a; +  «/)*. 

4.  9i 

8.    GV)"^. 

12.  3(a  +  25)-^ 

Change  each  of  the  following  radicals  into  expressions 
containing  exponents  in  the  form  of  fractions : 

13.  </^.  17.  ^/J+b.  21.    '-VT^^. 

14.  Va.  18.  V3  a-^.  22.   -^oT^. 

15.  V2^.  19.  -v/4  2^-4.  23.    ■^^^. 

16.  7aV^.  20.  ->/9^+VF^.  24.   </Qa  +  b)-^ 


260  ELEMENTARY   ALGEBRA        [Ch.  XVII,  §  215 

Free  each  of  the  following  expressions  from  negative 
exponents : 


25. 

a-2. 

26. 

Sa-i. 

27. 

9c-6. 

28. 

2  a-^b\ 

29. 

2c-5 

32.    ^"''-''^ 

30. 

16-2  a;- V. 

9x~^t/  "» 

4  a-^6^ 

33     25-2:r(a-J)-3 

31. 

2  a-i5-3c-* 
5x-2y-%-6 

4a~2(^a;— y)"* 

Find  the  value  of  each  of  the  following  expressions : 

34.  9l                       38.  3(49)i                42.  243"l 

35.  64*.                     39.  2(81)i               43.  256^. 

36.  (1002)i.                  40.  2.16"^.                   44.    64-*. 

37.  811                      41.  216"*.                   45.   (27  +  5)-*. 

Find  the  product  of  each  of  the  following  expressions . 

46.  a^  .  a-^  52.   (3  ^)-5  .  (3  ay. 

47.  X-''  •  x~'^^.  53.    5x  '  5  x-'^. 

48.  Sx-^^'4.aP.  54.   (a  +  by^-  (a  +  by. 

49.  a  .  ^-1.  55.   (-  ^)-^  .  (-  a)^+^. 

50.  5  6^  .  4  J-^.  56.   (-axy^-"" .  (- ao;)^-'^. 

51.  a'"-^  .  a3-2m^  57^    4^+1  .  22-m^ 

Find  the  quotient  of  each  of  the  following  expressions ; 

58.  b-^^b-^.  63.    a-2^-5^ -^  a-^^^-6^ 

59.  x-^-T-x.  64.   a-^^S^-^'^-^a^'^^-iJ*. 

60.  a~^  ^  a^.  65.    —  a-^S  a-^. 

61.  a-^^+i -^  ^-'^-1.  66,    7  ^-2-^532/ ^  ^7^^^-n^ 

62.  4:^;-^3rl-^  67.   8  ^-^5"^' -^  a^^J^^^'. 


Ch.  XVII,  §  216]        THEORY  OF  EXPONENTS 


261 


216.  The  index  laws  which  apply  to  monomials  apply- 
to  the  terms  of  a  polynomial. 

1.    Multiply  J—  a^+l  —  a"^  by  a*  +  1  —  a'K 

ai  —  a^  +  l-  oT^ 

a  —  a^+a^  — 1 

at  — a*+l—    a"^ 
—  a^  -I- 1  —     oT^  +  a^ 

a  —  a^  + 1  —  2  a~^  +  a~*  =  a  —  a*  + 1  -~  +  -^ 

In  this  chapter,  unless  the  contrary  is  stated,  results  are  to 
be  written  with  exponents  in  the  form  of  positive  integers  or 
positive  fractions. 


2.    Divide  x^\Jy~^—  2  +  x^'^V^  by  V^  V?/""-^  —  V^  ^v^. 


x^lf^  —  xy~^ 


xy~^  —  2  -{-x'^y^ 
xy"^'^  —  1 


-l+a;-y 
-1+0;-^^ 


1  _i        _i  1 
x^y  ^  —  x  ^?/3 


3  ,  1     _1  —11 

2/       2/^      ^^ 


3.   Extract  the  square  root  of  9:r— 12a:*+10  — 4:i;"'^+rr-i 

3  a;*  -  2  +  a;"^  .-- 


6  a;* -2 


9  a;    -12a;*  +  10-4a;""i  +  aj-i 
-12a;*+   4 


6  aji  —  4  +  a;"^ 


6  -  4  a;"*  +  i«"^ 
6  -  4  aj"*  +  x-^ 


3a;^-2  +  i. 


2()2  ELEMENTARY  ALGEBRA        [Cir.  XVII,  §  21G 

EXERCISE   CIX 

Find  the  products  of  the  following  expressions : 

1.  (x^  +  ybi^^-yb' 

2.  (x^  +  3  x) {x^  —  3  a;). 

3.  (a'^-i5  +  a^5-2-a^^-2J-i)a5. 

4.  (x+x-'^'-'l)(x—x-'^+V). 

5.  (x^  —  2/^)  (x^^  +  x^y^  +  y^). 

6.  (a;*  +  y^) (x'  ~  xy^  +  x^y  -  ^') . 

7.  (2a-2+3a-i-^5)(2a-2-3a-i+5). 

8.  (5  a;^-3/+3  _  2  rr^-y+i  -  xP-Y''^) 
(3  :r^+y-i  +  4  xP+Y~^  -  ^^^Y)' 

Find  the  quotients  of  the  following  expressions ; 
9.     (^x^''-y^'')-^(x''  +  y'').       12.     (a:2+2:-2-2)-^(>;-:r-l). 

10,  (a-3^-56^)~(a-^-62«).    13.    (^f  _2/^)_j.(^i_^i). 

11.  (ii;*-|-:r^- 6)^(2:*-2).   14.    (p^  -q)-^(p^-q^). 

15.  (a^-jt)^(a*+a*5i+J*). 

16.  (a;3__^2>^^(^J  +  ^i), 

17.  (2  a5J-3  -  5  a^}-^  +  7  a^J-i  -  5  ^2  +  2  aJ) 

18.  ( -v^^*  -  4  rry  +  4  y -v/^  +  4  2/2)  -^  ( -^^  +  2  x^y^  +  2 «/) . 

Extract  the  square  root  of  the  following  expressions : 

19.  x-^  —  Gx-^+llx-^—Gx+x^. 

20.  4a;"  +  9a;-"  +  28- 24a;  2_16a;^ 

21.  1  +4a:-^-  2a;~*-  4a;-i+  25 a;"^ -  24 a;"*  +  16 a;-2. 


Cii.  Xvn,  §  217]        THKORY  O^  EXPONENTS  263 

217.  By  the  principles  of  tlie  preceding  articles  many 
expressions  may  be  simplified. 

{a-b)-'  "■         \a     bj         ^         \  ab  ) 

=  -(a-b)\ 

2.    (a-'-^)"+^  +  -^^^  =  a"(--^)  +  -  =  a*^  -"  +  «  =  ^  +  a. 
^        ^  a  a  a" 

2"+^  •  2"-^  ""      2'^'      ""  2^' 

EXERCISE   ex 

Simplify  each  of  the  following  expressions,  giving  each 
result  in  a  form  free  from  radicals  and  from  negative 
exponents  : 


1. 


2. 


( 


■4a-'xh^  „     ^'"^^^"-^ 


8. 


■^a^6*c-i 


2*  •  9^  •  4^' 


8a2    N-i  ,«     2  a;2;y  ^     6« 


\27  a-^W 


Sa^-^     7^^i 


Va:a-i      a:<i'^  U.    (81  a*2;-2«/*3~*)i 


(-a)-^.a^T     •  '^'^^  ■  V.V  ^  :.i 


■    L     (-i)^     J 


x^i/^  ^  y_^z'^ 


7.    (64\/3t)  a*6-2)i  14.    (a;9-0^-(^''"0'' •(«''"*')'■■ 


264  ELEMENTARY  ALGEBRA        [Cii.  XVII,  §  211 


15. 


16. 


(a^y 


p-q 


■Va-^b^  ■  Va  •  V6-3 


a%-^ 


17. 


18. 


a-%\-^  _  /aS-iy 


a^-^ 


\[^^-VM 


19. 


20. 


21. 


22. 


2n+l  .  2^^ 


.  ^2(p-g)  I 
4^+1 


w+I 


(2^)^+1       (2^-^) 


3V&5       V5 


4Va269 


23.    0^—1  + 


i_l 


24. 


5. 


25. 


xi-1      x'-\-l      x^-l      2;^  +  l 


^^-    2&2-\(^5^j    •ll2JVaWyr 


27.  r(^¥'(^)"^i-^^-rcy*)-^°Cybn-^ 


.  XVII,  §  217]        THEORY   OF  EXPONENTS 
REVIEW   EXERCISE   CXI 

Simplify  the  following  expressions  : 

15.  a^. 

16.  ^4. 

17.  8ll 


265 


1.    V49 

5 


3.  a^6-"^ 

4.  ^144. 

5.  a/256. 

6.  a/8^. 

7.  v^aV^. 


29. 


30. 


18. 


3/ 


19.  Va2. 

4V6 

20.     -. 

3V2 


8.    Va2666'9. 


-  ^'^, 


10.  (3^1)2. 

11.  Ix^-GxK 

12.  VT8  •  V8. 


21. 

m   p 
(xn^q. 

22. 

{a  +  by. 

23. 

(2V^)2. 

24. 

V2  .  -^3. 

25. 

a/36  ^252. 

31. 


32. 


33. 


jam 

7V48 
3V27' 

a  +  x 


Va  +  o; 
V60  a%^ 


26.     V4a2^)%8. 


27.     (a2«+^^)3l^. 


13.  V32a2'^-iJ. 

14.  -y/^a-^Tb,      28.    (2  ax'^y^J^ 


V2  ^2^2 
34.     V^  .  VF". 

36.  VT2  •  ^7^. 

37.  V108-V72. 

38.  ■^54  +  -v'l28, 


39.  ■Vah-'^c-^  ■  (a-^-^c-^yK 

40.  (2-V'^5)(3  +  4V=^). 

42.    8~^  .  x^  .  ^a:V9^-f-(64  x"*)"*. 


43. 


1  ^  sfx^-  ~^}  -  C^^-:^-^). 


2GG 

44. 

45. 

46. 

47. 
48. 
49. 
50. 
51. 
52. 

53. 

54. 
55. 
56. 
57. 
58. 
59. 

60. 

61. 
62. 
63. 


a 

_2 


ELEMENTARY   ALGEBRA 
2 


1 

[Cii.  xvn,  §  2i; 


+  16*. 

Va"'  ■  Vo^^. 

5^5       3^20 

^2  .  -y  1  .  ^3. 

^2  .  </l .  </2. 

3VlT  _  7V22 
2V98         5 


(3Va-"'6»')-"'P. 
Va  +  J  •  -y/a—h. 
3^472  ^2-v'2^. 
V2i  +  V54-V6. 
2V3-V12+A/9. 
V20  +  VI5-V|-. 

\a%-y    '  [a-^bV  ' 
V75  +  V48-V243. 
(2  +  3V5)(3V5-2). 
2aJcV20  +  3aV5lV. 


64. 


65. 


V5  +  V'2 

V7  +  V2 
y+2Vi4' 


66.    Vi9_8V3. 

67 


68. 
69. 

70. 

71. 

72. 


V27-I2V5. 

V^  +  3 

V^-2V3' 

2^4  +  5v'32 


a/108 


^2(n— 3n 


-5m-» 


r,Sm-4n 


a'c 


-Wb 


</b^ 


73.    -Va^—x^  + 


a-^Vab^ 
x^ 


V«2- 


■X* 


74.  2V3+3V|-5V-i/. 

75.  (6x2-6)( 


^2a;  +  2y 
76.    4V|-fV^-2V27. 

1-3:2 


77. 


78.    2V24  .  3</18  .  4a^24. 


79.    Va*"J"c2p  .  ^a^bc^-P. 


H.  XVII,  §  217]        THEORY  OF  EXPONENTS  267 


80. 


81. 


82. 


84. 


94. 


95. 


^'-^  +  ^  \  86.    V22-flOV-3. 

87.    (_Vx''^</af)T=i. 

1  +  V3-V5"  „«     JJ.fA'^^ 


2V8-3 

+  V3-A 

V^=^-3V^ 


88.    a^t/      ^    J 


a;"'  x* 


V-2  +  2V-5  89.    (a^x-^-^-Va^x-^y\ 

x+1/-^       x-y-^  90     <^x"'Y"-"  .  /y"'+»Y 


V3        2-V^  -^ 


91. 


2-V3     2  +  V-2 


Va  +  VJ      Va  —  V6 " 


2^  1  92.    7\^54  +  ^256  +  ^432. 


85.    j4 


(l-x^y      (l-x^y         93.    (81a;-16)--(3a;*+2). 


V34-V5-V5-V5 

5,1  .      -,      3 


96.  (9aS-21  a3V^_2^5  a:^^  +  12a-l:r*)-^(f  a3-4ax*). 

97.  Arrange  in  order  of  magnitude  :   V|,  V |,  Vf . 

98.  Arrange  in  order  of  magnitude  :   (|)^  and  (f)*. 

99.  Extract    the   square  root   of    x^  +  ^xy^+y'^+A^x^y 
-2x^y''^-4:xi. 

.00.   (a-4  _  5  J4  +  4  ab^)  -  (a-3  +  2  a-^  +  3  a-iJ2  +  4  J3)^ 


CHAPTER   XVIII 

QUADRATIC  EQUATIONS 

218.  An  equation  which  contains,  in  its  simplified  fornic 
the  second  power  of  the  unknown  quantity  as  the  highest 
power  of  that  unknown  quantity  is  called  a  quadratic 
equation. 

Thus,  —  H —  =  f-  -,  which,  when  simplified,  becomes 

2  ic-  —  6  it*  —  1  ==  0,  is  a  quadratic  equation. 

219.  Every  quadratic  equation  may  be  reduced  by  the 
fundamental  laws  of  algebra  to  the  general  form, 

ax^"^bx  +  c  =  0, 

wherein  a,  J,  and  c  are  known  quantities,  and  wherein  a^O. 
If  a  =  0,  the  general  form  becomes  hx  +  e=^  0,  which  is  n 
simple  equation. 

If,  in  the  general  form,  &  =  0,  the  resulting  equation, 
aoi?  +  c  =  0,  is  called  an  incomplete  pure  quadratic  equation. 

If,  in  the  general  form,  c  =  0,  the  resulting  equation. 
ax^+  hx  =  0,  is  called  an  incomplete  quadratic  equation. 

If,  in  the  general  form,  neither  J  =  0,  nor  c  =  0,  the 
resulting  equation,  ax"^  -\-hx+  c=0^  is  called  a  complete 
Xor  affected)  quadratic  equation. 

The  known  numbers,  a,  6,  and  c,  are  called  the  coeffi- 
cients of  the  equation  ;  and  c  is  further  called  the  absolute, 
(or  Constant)  term. 

268 


Ch.  XVIII,  §  220]       QUADRATIC   EQUATIONS  269 

Thus,  oi^  =  4:y  or  0^  —  4  =  0,  is  an  incomplete  pure  quadratic 
equation  in  which  a  =  1,  6  =  0,  c  =  —  4 ;  3  a:;-  +  4  a;  =  0  is  an 
incomplete  quadratic  equation  in  which  the  coefficients  are 
«  =  3,  Z>  =  4,  c  =  0;  4a;^  +  4a;  +  3  =  0isa  complete,  or  affected, 
quadratic  equation  in  which  a  =  4,  6  =  4,  c  =  3. 

PURE   QUADRATIC   EQUATIONS 

^2.  (1) 

Extracting  the  square  roots  in  (1),  ±  x  =  ±  a,  (2) 

(+x=+a,  (3) 

The  complete  form  of  (2)  is 


—  x=  —  a,  (4) 

—  x=+a,  (5) 
x  =  —  a.  (6) 

A  value  of  —x  is  not  required;  therefore, 
multiplying  (4)  by  —  1,  x=      a,  (7) 

multiplying  (5)  by  —  1,  x  =  —  a.  (8) 

It  is  evident  that  (3)  and  (7)  are  indentical ;  and  that  (6) 
and  (8)  are  identical.  Hence,  if  the  double  sign  be  used  only  in 
the  right  membei^,  the  roots  are  not  altered  in  value.     Thus, 

Extracting  the  square  roots  in  (1),  x  =  ±  a. 

Verification  :  a^  =  a^, 

2.    Solve:  ^-20  =  ^.  (1) 

4  5  -^ 

(Clearing  of  fractions  in  (1),   5x^  —  400  =  4  x^,  (2) 

transposing  and  uniting  in  (2),  a?  =  400,  (3) 

extracting  square  roots  in  (3),  a;  =  ±  20. 

Vkhikication:  4p0_20  =  400_ 

4  5 

Note.  ,  If  x^  is  negative,  the  signs  of  all  terms  must  be  changed, 
since  the  square  root  of  a  negative  number  cannot  be  obtained. 


270  ELEMENTARY    ALGEBKA       [Ch.  XVIII,  §  22j 

EXERCISE   CXII 

Solve  the  following  equations : 
1.    ^2=169.  ^^     x+5  _  2x  +  7 


2.    x^-a^=0.  ^  +  1-3      3a;  +  18 

3 


3.    :?;2_81  =  0.  12.   f+^-^=-19i+: 


4.     3^:2=48. 


13. 


1       'T-l 


^+  9"     .     ^ 


5.  25x'^-b^  =  0.  x^y  x  +  ^     x^-l 

6.  a^x^=b^x^,  -.     r?:  +  a,a;— a      7 

14.    ^ -| =6. 

7.  lla;2=36  +  22:2.  ^""^      ^  +  '* 

,  8.    x^  =  a^+2ah  +  b^  15.   2^  =  ?^. 

(?:r  +  a      ax  —  0 

9.    ax2  —  ah  —  2  ax^.  ^        t\  i 

a{x  —  0)  _a_^bx 

10.    (7  2;)2=296-(5:i;)2.  •         6a;       ""6      a' 

17.   ^—2  + 2  =  3a*  +  l. 

12/        .   .      2    \         5 


-  f  (^->+4i) 


a;  +  l 


SOLUTION  OF   QUADRATIC   EQUATIONS   BY   FACTORING 

221.  If  the  product  of  two  quantities  be  zero,  either 
of  the  two  quantities  may  be  taken  as  equal  to  zero. 
When  the  left  member  of  a  quadratic  equation,  reduced 
to  the  general  form,  can  be  factored,  either  factor  may 
therefore  be  taken  equal  to  zero,  or  equated  to  zero.  The 
roots  of  the  factors  are  therefore  the  roots  of  tlie  equation. 

The  Factor  Method  holds  for  all  forms  of  quadratic  ; 
equations,  botli  complete  and  incomplete. 


Til.  XVIIT,  §  221]       QUADRATIC    EQUATIONS  271 

1.  Solve  by  factoring  :   9  x^=  36.  (1) 
Dividing  (1)  by  9,                                     x'=:4:,  (2) 

transposing  in  (2),  a^  —  4  =  0,  (3) 

factoring  in  (3),  (x  +  2)  (x-2)  =  0,  (4) 

equating  each  factor  in  (4)  to  zero,     \  '  (5) 

^.  X  ~^-~  ij  —  Uj 

transposing  in  (5),  x  =  —  2,  or  x  =  2.  (6) 

Verification:    9(-2/  =  36:  9(2)2=36. 

2.  Solve  by  factoring  :    ax^+  hx  =  0.  (1) 
Factoring  in  (1),                           x{ax  +  ?>)  =  0,  (2) 

^  (3) 

ax+'b  =  0, 

transposing  in  (3),  a;  =  0,  ax  =  —  b,  (4) 

dividing  ax  =  —  bhy  a,  x  = •  (6) 

Verificatiox  : 

a(0)+5(0)=0;  af -^^ +  bf --\=^^' --  =0. 
\     aj         \     (^J     <^t       ^ 

3.  Solve  by  factoring  :    x^-4:x-21  =  0.  (1) 

Factoring  in  (1),  (x  -  7)  (a;  +  3)  =  0,  (2) 

(-/^ 7  =  0 

equating  each  factor  in  (2)  to  zero,     /  ^  (3) 

\^  X  -j~  o  -^  ", 

transposing  in  (3),  x  =  7,  or  x  =  —  3.  (4) 

Vekification  : 

I  (7y-4(7)-21  =49-28-21=0. 

I    (_3)2_4(_3)_21=   9  +  12-21  =  0. 


272  ELEMENTARY   ALGEBRA     [Ch.  XVIII,  §§  222, 223 

EXERCISE   CXIII 

Solve  the  following  equations  by  factoring  : 

1.  2^2+7^+12  =  0.  8.   3:z:2_25^  +  28  =  0. 

2.  x^+x-m  =  0.  ^    15:^2+23a:-28  =  0. 

3.  ^2_^_i2  =  0.  ^^     -63a.2+16:?;-l  =  0. 

4.  :z;2  _(.  9  ^  +  20  =  0. 

5.  :^2+2a;-224  =  0.  ^^-   ^'"2!^  20^^* 

6.  :^2_7^_260=0.  1 

12.   4:0x^-x-  —  =0. 

7.  2:i;2+9a;-5  =  0.  20 

13.  x^  —  (a+h)x+ab=0, 

14.  :z;2-a;(2j9  +  5^)  +  10jt?^  =  0. 

NUMERICAL   COMPLETE   QUADRATIC   EQUATIONS 

222.  If  the  coefficients  of  the  equation  ax^-{-bx+c  =  0 
are  numerical,  the  equation  is  called  a  numerical  complete 
quadratic  equation. 

Thus,  5x^  +  7  X  — 3  =  0  is  a  numerical  complete  quadratic 
equation. 

223.  Solution  by  completing  the  Square.     By  §§  73  and 

74,  (ir  ±7^)2  =  x^±2nx+  n^.  The  third  term  is  evidently 
the  square  of  half  the  coefficient  of  x.  If  the  left  member 
of  a  complete  quadratic  equation  contains  the  unknowns 
only,  and  the  right  member  the  absolute  term,  the  equa- 
tion may  be  made  to  assume  the  form  x^  ±2  nx  by  dividing 
the  equation  by  the  coefficient  of  x"^.  The  left  member 
may  be  put  into  the  form  of  the  square  of  a  binomial  by 
adding  tlie  square  of  half  the  coefficient  of  x^  a  process 


t 


Ch.  XVIII,  §  223]       QUADRATIC   EQUATIONS  273 

which  is  called  completing  the  square.     This  process  is 
best  understood  hy  examples. 

1.  Solve  the  equation:  x^—6x=  16.  (1) 

The  left  member  is  already  in  the  form  3if—2nx]  that  is, 
the  coefficient  of  x^  is  unity.  Half  the  coefficient  of  x  is  —  3  ; 
(  —  3)^  =  9.  Therefore,  adding  9  to  both  members  of  (1),  so  as 
not  to  destroy  the  equality,  or, 

completing  the  square  in  (1),     x^  —  6x  +  9  =  2o,  (2) 

extracting  the  square  roots  in  (2),       a;  —  3  =  ±  5,  (3) 

transposing  and  uniting  in  (3),  it?  =  3  +  5, 

or,  ic  =  3— 5, - 

combining  in  (4),  x=8,  or  x  =  —  2.  (5) 

Verification  : 

(8)2_6(8)  =  16;  (-2)2_6(-2)  =  16.  • 

2.  Solve  the  equation :  a;^  _  14  ^  __  1][  _.  0.  (1) 
Transposing  in  (1),                       cc^  —  14  a;  =  11,  (2) 

completing  the  square  in  (2),  o?^  —  14  a;  +  49  =  60,  (3) 

extracting  the  square  roots  in  (3),        x  —  7  =  ±2  VT5,  (4) 

transposing  and  uniting  in  (4),  x  =  7  +  2  Vl5, 

or,  aj  =  7— 2V15. 


'}  (4) 


(5) 


IFICATION  : 

+  2V15)2  -  14(7  +  2Vl5)  -  11  = 

49  +  28  Vl5  +  60  -  98  -  28  Vl5  -  11  =  109  - 109  =  0. 

(7  -  2 Vl5)2  -  14(7  -  2Vl5)  -  11  = 
49  -  28Vi5  +  60  -  98  +  28v'15  -  11  =  109  - 109  =  0. 


274  ELEMENTARY   ALGEBRA       [Ch.  XVIII,  §  223 

3.  Solve  the  equation  :  x^—Sx=4:.  (1) 

Completing  the  square  in  (1),  a;^  —  3  a;  -f  f  =  4  +  |  =  ^^\  (2) 

extracting  the  square  roots  in  (2),          x  —  ^=±  |,  (3) 

transposing  and  uniting  in  (3),      x  =  4,  or  it^  =  —  1.  (4) 

Verification:     (4)2-3(4)  =  4;    (- 1)^- 3(- 1)=  4. 

4.  Solve  the  equation: 1 :!_==_. 

^  22:  +  !      3-2:      6 

Clearing  of  fractions  in  (1), 

5(3  -  x)  +  5(2x  +  l)=6(2x  +  1)(3  -  x), 
simplif3dng  in  (2), 

15  -  5  x  +  10  X  -{-  5  =-  12  x^  +  30  x  +  18y 
transposing  and  uniting  in  (3), 

12x'-^25x  =  -^2, 

dividing  (4)  by  12,        x^-^j^  =  -^, 

completing  the  square  in  (5), 

^_25_^'     /25Y^/25Y__  2      529 


12       V24y       V24y       12     576'  ^  ' 

extracting  square  roots  in  (6), 

^  2.5. L.    2  3  /7> 

transposing  and  uniting  in  (7),  x=2,  or  x  —  -f^,  (8) 

Verification  : 

1      j^6.  _1_,_1_^6     12^42^6 

5"^        5'J.  +  l"^3-iV     ^^'^     ^^^     ^'' 

Rule  for  solving  Numerical  Complete  Quadratic  Equations : 

After  clearing  the  equation  of  fractions  (if  any  exist)  ^  trans- 


Ch.  XVIII,  §  223]       QUADRATIC   EQUATIONS  275 

pose  the  unknow7i8  to  the  left  member  and  the  absolute  term 
to  the  right  member  ;  divide  the  equation  by  the  coefficient  of 
x^ ;  complete  the  square  by  adding  to  each  member  the  square 
of  half  the  coefficient  of  x ;  extract  the  square  root  of  each 
member;  solve  the  simple  equations  thus  derived, 

EXERCISE   CXIV 

Solve  the  following  equations  by  completing  the  square : 
'  17.    2  a;  +  14  +  -  =  0. 


2. 
3. 

a:2  +  12x=13. 

x'^  +  x-2--^  0. 

18. 

X 

4 

4. 
5. 

x^  +  x-^i^  0. 

:0. 

19. 

X"^         X 

9,  2a;      8      A  on     113.r— :^i'r2— _  411 

7.  x2  +  2x  +  40  =  0. 

8.  2:2-32:+l  =  0. 

9.  2^2  + 5^-7  =  0. 

10.  ?>x'^+bx=2, 

11.  3  2;2-7a;=16. 

12.  2a;2-52:+3  =  0. 

13.  x(x+ 1)^12. 

14.  a:  +  3  = 


21. 

2            13 
a;-l      x  +  3      8 

22. 

5         2_    14 

x+ 1     X     a;  +  4 

23. 

5                3          2 

4«2_i      2a;+l      3 

*>A 

a;  +  3a;-3      2a;- 3 

x-\-l      x-2~   x-\ 

25. 

x^+Q        3    _,        7 

«  a;2_4     2-ir  x  +  l 

15-    ^  +  "  =  2-  26.    9^a;2-90^a:=-19.5. 


2    ,    -  5  a;  „„    17      32-II.t 

h  4  = 27. — : 

3a;  62^  +  7  X  "6  x^ 


276  ELEMENTARY   ALGEBRA       [Ch.  XVIII,  §  224 

LITERAL   COMPLETE   QUADRATIC   EQUATIONS 

224.  If  the  coefficients  of  the  equation  ax^  -\-hx  +  c  =  () 
are  literal,  the  equation  is  called  a  literal  complete 
quadratic  equation. 

Thus,  2  ax^  +  mx  +  6  n  =  0  is  a  literal  complete  quadratic 
equation. 

The  solution  is  found  in  the  same  manner  as  in  the  pre- 
ceding paragraph. 

1.    Solve  the  equation :  x^  —  hx  —  cx=(a  +  l))(a  —  c).   (1) 
Factoring  in  (1)  to  show  coefficient  of  x, 

:^^x{h  +  c)  =  {a  +  h) (a-  c),  (2) 

completing  the  square  in  (2), 

;^-x(b  +  c)  +  (^^  =  (^±^+(a  +  h){a-c),     (3) 
^bplifying  the  right  member  in  (3), 


^._,(,+e)+(^j^J-*«^+^«*-^;''+'^- 

-2hG  +  G^ 

(4) 

extracting  square  roots  in  (4), 

""        2    ~^         2        ' 

(5) 

transposing  and  uniting  in  (5), 

x  =  a-\-h,  ov  x  =  c  —  a. 

(6) 

Verification  : 


(  (a+by-(a+b)(b-^c)  =  (a+b)(a-c), 
(a-\-b){a  —  c)  =  {a-{-b){a  —  c), 

(c-a)"-(c-a)(b-\-c)  =  (a-\-b){a-c), 
(c-a)(-a-5)  =  (a  +  6)(a-c). 


Ch.  XVIIT,  §  224]       QUADRATIC   EQUATIONS  277 

2.    Solve  the  equation  :  ax^  -{-  be  —  hx  =  acz.  (1) 

Transposing  in  (1),  aaj^  —bx  —  acx  =  —  be,  (2) 

factoring  in  (2),  ax^  —  x(b  +  ac)=^  —  be,  (3) 

dividing  (3)  by  a,     a^  -  x  f^+^^\  =  -  -,  (4) 

\     a     J  a 

completing  the  square  in  (4), 
simplifying  the  right  member  in  (5), 

extracting  square  roots  in  (6), 

^_b  +  ac^     b-m  ,^. 

2a  2a   '  ^  ^ 

transposing  and  uniting  in  (7),         x  =  -,  ov  x  =  c.  (8) 

Qj 

Verification^  : 

.     \ay  \aj         \aj    a  a 

ac^  +  bG  —  bc  —  a(?. 

The  left  member  should  always  be  factored  to  show  the 
coefficients  of  :jfi  and  of  x, 

EXERCISE   CXV 

Solve  the  following  equations  by  completing  the  square  i 

1.  a;2  +  4  J:r  =  —  4  J2.  4.    ^ :i^—%'pq=z'i'pq~Zqx. 

2.  2;2  —  5  aa:  +  6  ^2  =  0.  5.    hx^  +  ac  =  (a  +  bc}x. 

3.  x^  +  ax-'2a^  =  0.  6.   x'^  +  Qa +  h)x  + ab  =  0. 


278  ELEMENTARY   ALGEBRA       [Cii.  XVIII,  §  224 


7.  x^  +  ax  =  €?, 

1  ,   1 

8.  T  +  -  =  ^  +  — 

X  a 

9.  Ix^  +  mx  +  ^i  =  0. 
10.   x^  —  lax  +  h^^. 


20 


x—h      X-—  a 

21.  :r  (1  —  x^  =  ax^  +  b. 

22.  1  —  7  x/^=2ax  —  5:r^. 

11.  ^LZl^_^+^_5                   ^^  (^_a)2^         :r  +  a 

12.  ax'^  —  2bx-i'C  =  0.  x  —  a 

^                                24.  -  + =  a:  +  -. 

14.  ^  ^  >T  +  ^ 

^                                             25.  a22'-2/>2=^J.^-±_^. 

15.  aa;2+a  =  (^2_|_i^^^  a;  +  l 

16.  x^-lax-Va^-^W-^^.        2g^  ^1       &     ^     2a 

*  J  +  a;      a  +  2^      2a-/> 

17.  -1 — ^=^±4- 

a-x      a  +  x      a^-x^  ^_a      x-2h          b    ^ 

27, =  — — • 

18.    mqx^  —  mnx  +  pqx  =  np.  ^            x—  o       a  +  o 

4^2^j2              -^-  ^+2              2a;-l 
a-\'X      a  —  x__^bQa-\-b^ 


30 


a  —  J  +  2.T  a  +  S  —  a^ 

31.  cx'^-{a  +  b-\-c)x-\-{a-\-b^==0. 

32.  mTz:?;^  — (m  +  ?i)(m7Z  +  l)x^ +(m  +  7i)2=  0. 

33.  2x^a^-b'^)--{?>a^  +  b'^)(x-l)=^{W^^a^^(x-\-r) 


Cn.  XVIII,  §  224]       QUADRATIC   EQUATIONS 


2Yi) 


34. 


+ 


h  +  X       X—  a      2(a  +  5) 
35.   x\a  4-  5)2  -  x(a2  -  J2)  =  ah. 


36. 


37. 


2(a  +  ^>)  .     2  5    ^3(^-5) 
X  —  h         X  —  a       ir  —  35 

a:— ^  X     _     2(x  +  g) 


x-^^h      x  —  h      x+  ^a  —  4:b 


33^    ajl-^x^)  ^   (2a-b)x^    ia 
hx  a  a  +  b 


39.   \(x+l)  +  -(x-\^ 


2a^-l 


40.     {n  —  x)(\- 


Za  +  Zx 


c-h 


y^A 


X 

l-2a 


■  2c 


(c-3)- 


l+g 
1-ic 


41. 


x+h  ,     2a 


X—  0  0 


2a -b\ 
X  —  b  J 


42.    (a2  +  62)(4 a;2  +  1)  +  2 a5 (4 x^-l)  =  'ix(a^~  b^). 
4 


43.    ax 


ax  +  b 


lb\l+x)x-a\l-x')'\  =  b. 


^  ^a;-l      a  +  lV       a;     a(a;-l)y 

2  rt(a  +  5)  -  b^x  2 


45 


46. 


Jx—  2  a 


lf^-2^      " 
b\2a       J 


b_n  _  _y 

a^\x     2  a, 


2x] 


r    4a 


il-i      1_1 
.a;       a      X      b_ 


=  0. 


280  ELEMENTARY  ALGEBRA      [Ch.  XVIII,  §  225 

SOLUTION   OF   QUADRATIC   EQUATIONS   Bf   A  FORMULA 

225.    Every  quadratic  equation  may  be  reduced  to  the 
general  form,  ax^  +  hx  +  c=^Q. 

Solve  the  equation  :  ax'^  +  hx  +  c  =  Q.  (1) 

Transposing  in  (1),  ax^  -f  6a;  =  —  c,  (2) 

dividing  (2)  by  a,  ^  +  ^Q  =  _  £,  (3) 

completing  the  square  in  (3), 

\aj     4  a"^     ^a^     a 


simplifying  the  right  member  in  (4), 

\al     Aa^         4a2 
extracting  square  roots  in  (5), 


'+r„=*^^^'       ('') 


transposing  and  unitmg  m  (6),         jr  = ^ y 


x  =  — 


a) 


2a 

The  values  of  x  in  (7)  are  general  values.  The  values 
of  the  roots  in  any  particular  equation  are  found  by  sub- 
stituting in  the  formulas  in  (7)  the  values  of  a,  6,  and  c 
in  any  particular  equation. 

1.    Solve  by  the  formula  :   2  a;^  --  5  ir  =  3.  (1) 

Putting  (1)  in  the  general  form,  2  a?^  —  5  a?  —  3  =  0.  (2) 

In  (2),  a  =  2,   b  =  -5,   c=-3.  (3) 


(  11.  XVIII,  §  225]       QUADnATlC    EQUATIONS 


281 


Substitute  for  a,  h,  and  c  their  values  from  (3)  in  the  formu 
las  in  (7), 


5+V(-5r-4(2)(-3)^3 
2(2) 


or, 


_^_5-V(-5y-4(2)(-3)_      , 
2(2)  ■''■ 


(4) 


Verification  : 


2(3)2  -  5(3)  -  3  =  18  -  15  -  3  =  0, 


.2(-i)^-5(-i)-3  =  i  +  |-3  =  0. 
The  formulas  are  written  in  the  more  compact  form, 


x- 


2a 


EXERCISE   CXVI 

Solve  the  following  equations  by  the  formula : 
a^  -  10  a;  +  25  =  0. 

5a;2+ii^  +  83  =  0. 
<a;  -  2)  =  5(x  -  6)2.  8.    ax^-2bx  +  c  =  0. 

9.    aa?  —  (a  +  b')x  +  6  =  0. 
2x(2x-b')  2 


5.  23+3^=io|. 

6.  <.t.  +  l)  =  W- 

7.  x'^  —  2ax  +  b  =  (i. 


10. 


=  3. 


2a;-l  2a;-l 

11.  (a  +  J)  6a;2  +  a^  =  a(a  +  2  J)a;. 

12.  (a;  -  2)2  +  (a;  +  5)2  =  (a;  +  6)2. 

13.  (J2_l)2;2_2(a6_l)x  +  a2=l. 


14.    (a  -  l)a;2  +  (a  +  l)a:  • 


a-1 


:0. 


282  eleme:ntary  algebra        [Ch.  xviii,  §  220 

IRRATIONAL   QUADRATIC   EQUATIONS 

226.  Quadratic  equations  which  contain  indicated  roots 
of  the  unknown  quantities  are  called  irrational,  or  radical 
quadratic  equations. 


Thus,  V.^' -{-3  +x  =  9,  is  an  irrational  quadratic  equation. 

Roots  obtained  in  solving  quadratic  equations  involvinc/ 
radicals  must  he  substituted  in  the  original  equation  for  the 
purpose  of  verification. 


1.    Solve  the  equation  :        V:r+ 1  =  x-\-l,  (1) 

Squaring  (1),                              ■  x  +  1  =x'  +  2  x-\-l,  (2) 

transposing  and  uniting  in  (2),  ^  +  x  —  ij  =  0,  (3) 

solving  (3),                                                a;  =  2,  or  a;  =  —  3.  (4) 

I  -s/2  +  l  =  2  +1 ;  2  is  a  root  of  (1). 
Verification:  ^  ^  ^ 


{  V—  3  +  7:^  —  3  +  1;  —  3  is  not  a  root  of  (1). 


2.    Solve  the  equation  :   V2  a;  +  21  -  Vx+1  =  2.       (1) 
Squaring  (1), 


2a;  +  21-2V2ic2  +  35a;  +  147  +  i»  +  7  =  4,  (2) 

transposing  and  uniting  in  (2), 


-  2  V2  ar^  +  35  aj  + 147  =  -  24  -  3  a?,  (3) 

squaring  (3),  8  aj^  +  140  a?  -f-  588  =  576 + 144  a?  +  9  x\    (4) 

transposing  and  uniting  in  (4), 

aj2_j.4a;-12  =  0,  (5) 

solving  (5),  a^  =  2,  or  x  =  —  Q>.  (6) 

f  V4T2i  -  V2T7  =  2  ;  2  is  a  root  of  (1). 

VERIFICATION :    \        ^    ^ 

^  V- 12+21- V-0+7=2;  --G  is  a  root  of  (1) 


Ch.  XVIII,  §  220]       QUADUATIC    EQUATIONS  283 

EXERCISE   CXVII 

Solve  the  following  equations  and  verify  the  roots : 


„       Vx  +1  —  2        o    /-      ^ 

3.  y'x  +  5  +  x  =  7.  7.  ;.  =2V^+1. 

2Va:;—  1 

4.  4  a;  —  Va;  +  3  =  a;  —  5. 


5.    9x--\/'^x  +  l  =  2x-l. 


'    a;  —  1  X 


9.    Va;  +  1  +  V5(a;  +  2)  =  3. 
10.    V2a;-7  +  V7¥+8  =  ll. 


11.  VB  a;  +  4  +V5(x  +  1)  =  9. 

12.  Va  +  X  +  V6  —  x  =  Va  +  6. 


13.    2V3¥+l-3V^H^  +  2  =  0. 


14.  3V3x-4  +  4x=10(a;-l). 

15.  V5^-2x+l^v^^3-3. 

2Va; 

16.  gVx+S- Vi^=2V2  3;+2. 


17.  3V3a;+l-2Vx+3=V2(a;+l). 

18.  y/Tx+l  +  3V9a;-2  =  5Vox-l. 
a;  ,  a;  8  a 


19 


Vx+Va— a;      Vx— Va— a;      3Va; 


284  ELEMENTARY  ALGEBRA        [Ch.  XVIII,  §  227 

SOLUTIONS  OF   EQUATIONS  IN  THE   QUADRATIC   FORM 

227.  An  equation  which  contains  only  two  different 
powers  of  the  unknown  quantity  one  of  which  is  double 
the  other  is  said  to  be  in  the  quadratic  form.  The  general 
type  of  equations  in  the  quadratic  form  is  ax^"  +  bx''  +  c  =  0. 
Equations  in  the  quadratic  form  may  be  solved  like 
quadratics. 

1.  Solve  the  equation :  x^ -2x^ +  1==Q.  (1) 
Writing  (1)  in  the  quadratic  form, 

(aj2)2-20xj2)+l=O,  (2) 

factoring  (2),  {x^  - 1)  (x"  - 1)  =  0,  (3) 

equating  the  factors  in  (3)  to  zero, 

W_i=o,  ^^ 

transposing  in  (4),  a?  =  1,  x^  =  l,  (5) 

extracting  square  roots  in  (5), 

a:  =  ±  1,  or  a;  =  ±  1.  (6) 

Verification  :  1  —  2  +  1  =  0. 

2.  Solve  the  equation:  a;^  —  9 a?^  +  8  =  0.  (1) 
Writing  (1)  in  the  quadratic  form, 

(a;5)2-9(aj^)  +  8=0,  (2) 

factoring  in  (2),  {x^  -  8)  {x^  -  1)  =  0,  (3) 

equating  the  factors  in  (3)  to  zero, 

transposing  in  (4),  x^  =  8,  x"^  —  1, 


Cii.  XVIII,  §  227]       QUADRATIC    EQUATIONS  285 

raising  each  equation  in  (5)  to  |  power, 

{x')i  =  sK  (xi)^^(l)i  (6) 

simplifying  in  (6),             x  =  16,  or  x  =  1.  (7) 


Verification  : 


(16)"2  -  9(16)^  +  8  =  64  -  72  +  8  =  0, 
1-9  +  8  =  0. 


3.    Solve  the  equation  : 

x^-lx-  Vx'-lx  +  lS  =  12.  (1) 

Adding  18  to  each  member  in  (1), 


(a;2  _  7  a:  +  18)  -  Va;2  -  7  X  + 18  =  30,  (2) 

writing  (2)  in  the  quadratic  form, 


(Va;2-7:^  +  18)2-Vi»'-7a;  +  18  =  30,  (3) 

transposing  in  (3), 


(^/af  -7  X  +  lSy  --Vx'  -T  X  +  1S  -  30  =  0,  (4) 

factoring  in  (4), 


( Va;'  -  7  a;  +  18  -  6)  ( Va^'  -  7  a;  + 1 8  +  5)  =  0,  (5) 

equating  the  factors  in  (5)  to  zero. 


transposing  in  (6), 


-^ay'-7x  +  lS-6  =  0, 
Vi^  -  7  a;  + 18  +  5  =  0, 


Solving  Vaj2  _  7  a;  +  18  =  6,  x  =  9,  or  -2. 


(6) 


■y/x^-7x-}-lH  =  6,  Vaj2-7a;  +  18=-5.  (7) 


Var^  —  7  a;  +  18  =  —  5  is  impossible  since  the  radical  cannot 
equal  a  negative  quantity. 

Vkkificatiox:   on  substitution  in  (1),  both  9  and  —2  are 
roots. 


286 


ELEMENTARY   ALGEBRA        [Ch.  XVIII,  §  227 


EXERCISE   CXVIII 

Solve  the  following  equations : 

2.  a;4_5^__i26  =  0.  5.    x^-6xi  =  lG. 

3.  x*-30it^  +  125  =  0.  6.   2x-^-x-i-6  =  0. 

7.     :.-3_     -3^_    7 


^     —  X     —         g^ 

8.    {x  +  -]  —x—l2  =  -' 
\        xj  X 


9. 


55 


ix  +  iy    (^  +  7)    11 


10.  ^x^-d^  =  '2 » 

11.  V^T12+a/^"T12  =  6. 

12.  a;  +  Va;2  —  ao;  +  J'^  = \-h. 

a 


13 


3?  +  %X+U         X+4: 

14.  A/^^n[  +  2  v^^=n:  - 1  =  0. 


15.  a;2-2Va^  +  4a;-5  =  13-4a;. 

16.  49a:2  +  42a;4.9  =  l_(7a;-|-3). 

17.  3  a^  +  15  ^  -  2  Va^  +  5  x  +  1  =  2. 

18.  2a^  +  3a;-5V2a?  +  3a;  +  9  =  -3. 
3  2 


19 


-  +  1  =  0. 

(a^  _  5  iP  +  7)2      a^  _  5  a;  +  7      3 


20.  4a;2  +  22.'r-3V2a;2  +  lla;  +  13  =  78. 

21.  -\/x-^  +  ox+2S  +  </z^  +  5x  +  28-Q  =  0. 


Cii.  XVIII,  §  228]       QUADRATIC    EQUATIONS  287 

SOLUTIONS   OF   CERTAIN   HIGHER   EQUATIONS   BY 
QUADRATIC   METHODS 

228.  An  equation  wliicli  in  its  simplest  form  contains 
higher  than  the  second  power  of  the  unknown  quantity  is, 
in  general,  beyond  the  province  of  this  book.  Some  forms 
of  higher  equations  have  been  solved  in  the  preceding 
paragraph. 

1.  Solve  the  equation  :  4  ar^  +  8  a;2  __  140  a;  =  0.  (1) 
Factoring  (1),  4.x(a^ +  2  x-35)  =  0,  (2) 

equating  the  factors  in  (2)  to  zero, 

r  4a;  =  0,  ,3. 

solving  the  equation  in  (3)  separately, 

4i»  =  0,     x^-h2x-So  =  0, 
x  =  0.     (x+7)(x-5)  =  0, 

X  =  5,  or  a;  =  —  7. 
0  =  0. 
Verification  :     «  4  (5)^  +  8  (5)^  -  140  (5)  =  0. 

4(_7)-3_^8(-7)2-140(-7)  =  0 

2.  Solve  the  equation  :  :r^  =  21  a;—  20.  (1) 
Transposing  in  (1),  x'  -  21  a;  +  20  =  0.  (2) 
By  §  96,  a;  -  1  is  a  factor  of  x^  -  21  a;  +  20. 

Factoring  (2),   (x-l)(a^  +  x-20)=0,  (3) 

writing  (3)  in  prime  factors, 

(x-l)(x-A)(x+5)=:0,  (4) 

from  (4),  X  =  1,  or  x  =  4,  or  a:  =  —  5.  •  (6) 

Vekification:  1=21-20;  64  =  84-20;  -125  =-105 -20. 


288  ELEMENTAKY   ALGEBRA        [Ch.  XVIII,  §  228 

3.    Solve  the  equation  :  X* 4-4  a;3  + 2  :z;2- 4  r^:- 3  =  0.     (1) 

The  left  member  in  (1)  is  found  by  trial  to  be  the  square  of 
the  trinomial  x^  -\-2  x  —  1  if  4  be  added  to  that  member. 
Adding  4  to  each  member  in  (1), 

x'-^4.or^  +  2x^-^x  +  l  =  4.,  (2) 

extracting  the  square  roots  in  (2),  x^-{-2  x—l=  ±2,  (3) 

whence,  solving  (3),  x=l,  ov  x  =  ~  1,  or  x  =  —  3.  (4) 

1-1  +  4  +  2-4-3  =  0. 
Verification  :     il— 4  +  2  +  4  —  3  =  0. 

181-108  +  18  +  12-3  =  0. 

KoTE.  It  should  be  noticed  that  the  methods  of  solution  shown 
above  apply  only  to  pai-ticular  forujs  of  higher  equations  and  are  in  no 
sense  general  solutions  of  such  equations. 


EXERCISE   CXIX 

Solve  the  following  equations  : 

1.  a;(a;  +  2)(a;2-4)  =  0.  4.  x^  +  x^  +  x  +  l  =  0. 

2.  x(x  +  a}(x'^-P)  =  0.  5.  Sa^  +  4cx^-6x=7. 

3.  ax(ibx-2)(x'^-9)  =  0.  6.  afi-6  x'^-iS  x  +  50  =  0, 

7.  <2^-4)  +  (2;-2)  =  0. 

8.  Sa^-8x^  +  Sx+2=^0. 

9.  (r?;-l)(a;  +  2)(:2;2_6^+9)^0. 

10.  x^-2x^-x  +  2  =  0. 

11.  x^-10a^  +  nx'^  +  60x-64=0. 

12.  4:x^-12a^  +  n  x'^-12x-5  =  0. 

13.  Tx^-5a?-X^'>7x^  +  5(x+80)  =  0, 

14.  a^-Sx^-15a?  +  85x^  +  54:x-12  =  0. 


Ch.  XVIII,  §  229]       QUADRATIC   EQUATIONS  289 

CHARACTER   OF  THE   ROOTS 
229.    The  roots    of   the   general    form    ax^  +  bx  +  c=0 
have    been    found,    S  225,    to    be:    x  = ; — ^^^-^ 


Upon  the  nature  of  '\/}fi—^ac  will  depend  the  character 
of  the  roots.  The  quantity,  6-  —  4  ac,  is  called  the 
discriminant. 

(1)  If  6^  — 4ac  is  positive,  that  is,  if  J^— 4a(?>0,  the 
roots  are  real  and  unequal,  and  either  (jx)  rational  or  (6) 
irrational.  If  the  discriminant  is  {a)  a  perfect  square, 
the  roots  are  real  and  rational ;  if  (&)  not  a  perfect 
square,  the  roots  are  real  and  irrational. 

Thus,  in  the  equation  6  a?-  +  5  ic  —  21  =  0,  since  a=^%^  6=5, 
c=  —21,  the  discciminant  is  529  =  23^.  Therefore,  the  roots 
are  real,  rational,  and  unequal. 

In  the  equation  2a;^4-5a:  —  4  =  0,  since  a  =  2,  6  =  5, 
c  =  —  4,  the  discriminant  is  57.  Therefore,  the  roots  are  real, 
irrational,  and  un  jqual. 

(2)  If  6^  —  4  ac  is  zero,  that  is,  if  J^  =  4  ac^  the  roots  are 
real,  rational,  and  equal. 

Thus,  in  the  equation  4  a;^  —  12  a:;  +  9  =  0,  since  a  =  4, 
6  =  —  12,  c  =  9,  the  discriminant  is  0.  Therefore,  the  roots 
are  real,  rational,  and  equal. 

(3)  If  6^— 4a(?  is  negative,  that  is,  if  IP'—^acK^^  the 
roots  are  imaginary  and  unequal. 

Thus,  in  the  equation  a^  —  2aj  +  4  =  0,  since  a  =  1,  ?>=  —  2, 
and  c  =  4,  the  discriminant  is  —  12.  Therefore,  the  roots  are 
imaginary  and  unequal. 

The  character  of  the  roots  of  any  given  equation  may 
therefore  be  found  by  evaluating  the  discriminant. 


290  ELEMENTARY   ALGEBKxV       [Ch.  XVIII,  §  229 

The  following  summary  will  be  found  useful  : 

(1)  If  b-  —  iac> 0,  the  roots  are  real  and  unequal. 

(2)  If  b^  =  i  ac,  the  roots  are  real  and  equal. 

(3)  If  6-  —  4  ac  <  0,  the  roots  are  imaginary  and  unequal. 

1.  Determine,    without   solving,  the   character   of   the 
roots  of  2x^-lx  +  5  =  0. 

a  =  2,b  =  —  7,  c  =  5. 
b'-4.ac  =  A9  -  4(2)(5)  =  9  =  31 
Eoots  are  real,  rational,  and  unequal. 

2.  Determine,    without   solving,  the   character  of   the 
roots  of  9:i;2_i2a:  +  4  =  0. 

a  =  9,  6  =  -  12,  c  =  4. 

b'-4.ac  =  144  -  4(9)  (4)  =0. 

Eoots  are  real  and  equal. 

3.  Determine,    without   solving,  the   character   of  the 
roots  of  4  x^-  4  a;  +  5  =  0. 

a  =  4,  ?>  =  —  4,  c  =  5. 
52  _  4  ^^c  =  16-4(4)  (5)  =  -^  64. 
Koots  are  imaginary  and  unequal. 

4.  For  what  value  of  m  are  the  roots  equal  in  the  equa- 
tion 8x^+4x+  m  =  0  ? 

a  =  3,  6  =  4,  c  =  m. 

If  the  roots  are  equal,      ¥  —  4  ac  =  0, 

16-4(3)m  =  0, 

16  - 12  m  =  0, 

12  m  =  16, 

m  =  |. 


Ch.  XVIII,  §  230]       QUADRATIC    EQUATIONS  291 

EXERCISE    CXX 

Determine  by  the  use  of  the  discriminant  the  character 
of  the  roots  in  the  following  equations  : 

1.  x^-4:X+4:  =  0.  8.  a;2-7:i;+12  =  0. 

2.  x^-5x+Q  =  0.  9.  3x^-ix+l  =  0. 

3.  ^2_2:^;_1  =  0.  10.  2:^:2-13:r+5  =  0. 

4.  2a:2-3a;+5  =  0.  11.  3  2^2  -  4  2:+ 12  =  0. 

5.  5:i:2_2a;+l  =  0.  12.  2it2_5^_5  =  0. 

6.  ^2_3^_,_1^0.  13.  8x^~5x=2, 

7.  2^2-4  2;+ T  =  0.  14.  a;2__2ax=(J+a)(S-a). 

Determine  the  value  of  w  for  which  the  roots  are  equal 
in  the  following  equations  : 

15.  2x^  +  4:x+m  =  0.  18.    16x^+8mx+l  =  0. 

16.  mx^  +  Qx+S  =  0.  19.    4:x^—12x+m  =  0. 

17.  8x^+4:x— m  =  0.  20.    ma;2— (8  +  m):?:^  +  9  =  0. 

RELATION   BETWEEN   ROOTS  AND   COEFFICIENTS 

230.  It  is  convenient  to  derive  the  formula  for  the 
general  equation  ax^  +  bx+  c  =  0^  where  the  coefficient 
of    x^   is   unity.     Dividing   the    general   equation    by    a, 

x^-\ f-  -  ==  0  ;    in  the  last  equation,  putting  />  =  -,    and 

a      a  a 

q  =  -^  the  equation  is  x^  +  px  +  q  =  0. 
a 

The  roots  of  x^-{  px+  q  =  0  are  found  to  be 


292  ELEMENTARY   ALGEBRA      [Ch.  XVIII,  §  231 


Let  ^■^-p+Vj>^-4g^  (1) 


and  ^^-V--/¥^^^^  (2) 

adding  (1)  and  (2),        « +  ^  =  -j),  (3) 

multiplying  (1)  and  (2),    a^  =  q,  (4) 
Hence,  in  the  equation  x^+  px  +  q=0  : 

(1)  The  sum  of  the  roots  equals  the  coefficient  of  x  with 
its  sign  changed, 

(2)  The  product  of  the  roots  equals  the  absolute  term, 

231.  Since  the  equation  x^  +  px  +  q=Q  is  the  general 
form  of  complete  quadratics,  the  sum  and  the  product 
of  the  roots  of  any  complete  quadratic  may  be  found  by 
inspection. 

1.  Find  by  inspection  the  sum  and  product  of  the  roots 
of '2:^2  +  3^_,_l  =  0.  (1) 

Dividing  (1)  by  2/  a?  +  ^x  +  \  =  (),  (2) 

if  a  and  /3  are  the  roots,  by  the  rule,       a  -f-  y^  =  —  f ,  (3) 

a^  =  \,  (4) 

The  equation  x^  +  px  +  q  =  0^  wherein  a  and  y8  (read 
respectively,  '-'-  alpha  "  and  "  beta  ")  are  the  roots,  may  be 
written,  jr^  -  (a  +  p)jr  +  ap  =  0. 

2.  Form  an  equation  whose  roots  are  —  2  and  3. 
Take  oe  =  -  2,  and  ^  =  3. 

Then,     «  +  ^  =  -2  +  3  =  l;  ,x^  =  (- 2)  (3)  = -6. 
Substituting  for  «  +  j8 and  a/?  these  values  in  x^—{a-\-P)x+a^^^^ 
a;--(l)a;-6  =  0,  or,  x'-x-^-^O, 


CiT.  XVIIT,  §  231]       QUADRATIC   EQUATIONS  293 

3.  Form  an  equation  whose  roots  shall  be  the  squares 
of  the  roots  of  the  equation  x^  +  px-\-  q  =  0.  (1) 

Let  a  and  ^  be  the  roots  of  (1). 

By  the  conditions  the  required  equation  is, 

x'--.(a'  +  /3')x  +  a'^'  =  0.  (2) 

Now  a/3  =  q]    hence,  a^^^  =  (f. 

Again,  a  -f  y8  =  — /) ;   hence,  oC-  +  2a^-\-  ^^  =p^, 

a2  +  ^2^p2_2g. 

Substituting  a^  +  ^''=p^-2q  and  a^p''  =  q',  in  (2), 
x^-{lf-2q)x  +  q'=:z0. 

4.  Form  an  equation  whose  roots  are  reciprocals  of  the 
roots  of  the  equation  x^  +  px  +  q=0,  (1) 

Let  a  and  y8  be  the  roots  of  (1). 

By  the  conditions,  the  required  equation  is, 

\a      (3 J         a/3  \   a/S  J         a^ 

Now,  a  +  p  =  —p)  and  aji  =  q. 

Substituting  a  +  /3  =  —p,  and  a^^q,  in  (2), 

q        q       ' 
qa^+px +  1  =  0. 

The  results  obtained  in  the  preceding  examples  may  be 
verified  by  solving  the  equation. 


294  ELEMENTAHY   ALGEBRA        [Ch.  XYIII,  §  231 

EXERCISE   CXXI 

Form  that  equation  whose  roots  are  respectively; 


1. 

2  and  3. 

6.  4  and  —  |. 

2. 
3. 
4. 
5. 

5  and  2. 

6  and  —  2. 

—  3  and  —  6. 
I'  and  J. 

7.  -  5  and  |. 
8.-4  and  -|. 

9.  2  +  V3  and  2  -  VS. 

XO.  1+^^andl-A 

2         2 

XI.  -1 

-2V5 
3 

,   1  +  2V5 
and     3   . 

X2.  1  + 

2V^ 

-3andl-2V-3. 

13.  2+V-2and  2-V-2. 

14.  a  +  V&  and  a  —  VJ. 

15.  a  +  V  —  6  and  a  —  V—b. 


16.    —  (?  +  V—  d  and  —  ^  —  V  —  c?. 

17.  Form  a  quadratic  equation  whose  second  member 
shall  be  0,  whose  absolute  term  in  the  first  member  shall 
be  —  4,  and  one  of  whose  roots  shall  be  —  i. 

18.  One  root  of  the  equation  4a:;2__ig^_l_4_0  is 
2  +  V3  :    find  the  second  root. 

19.  Find,  without  solving,  the  sum  and  product  of  the 
roots  of  the  equation  8x^  —  lx— 5  =  0. 

20.  Form  an  equation  whose  roots  shall  be  the  recip- 
rocals of  the  roots  of  the  equation  2  a;^  _  ^  _)_  j  ._  q^ 

21.  Form  an  equation  whose  roots  shall  have  the  same 
absolute  value  as,  but  signs  opposite  to,  the  roots  of 
x^+px  +  q=0. 


Cii.  XVIII,  §§  232,  233]      QUADRATIC   EQUATIONS 


295 


GRAPHS   OF   EQUATIONS   OF  THE   SECOND   DEGREE 

232.  By  the  method  employed  in  §  145  it  is  possible  to 
construct  the  graph  of  any  equation  of  the  second  degree 
ill  two  unknowns. 

Consider  the  equation  y  =  ao?  +  hx  +  (?, 

the  right  hand  member  of  which  is  evidently  a  part  of 
the  general  form  of  complete  quadratic  equation  in  one 
unknown.  The  graphs  of  certain  numerical  forms  of 
y  =  ao?  -{-hx  +  c  for  various  characters  of  the  roots  are 
interesting. 


233.   (1)  When  52>4  qq^  and  when  Vi^— 4  ac  is  rational. 
1.    Plot  the  graph  ofy=2?  —  42:+8. 


Pi 

P2 

P. 

A 

P. 

P. 

Pr 

Ps 

P9 

x= 

—2 

-1 

0 

1 

2 

3 

4 

5 

6 

y= 

15 

8 

3 

0 

-1 

0 

3 

8 

15 

In  the  table  are  found  the  coordi- 
nates of  the  various  points.  Locat- 
ing convenient  points,  and  drawing 
a  smooth  curve  through  these  points, 
the  curve  P,P,P,P,P,P,P,P,P,, 
Fig.  12,  is  the  graph.  The  graph  is 
Rpen  to  cut  the  X-axis  at  the  points 
1\  and  P(;,  whose  coordinates  are 
respectively  (1,  0)  and  (3, 0).  But, 
since  the  ^/-coordinates  of  the  points 
where  the  graph  crosses  the  X-axis 
are  zero,  the  aj-coordinates  of  these 


-\ 

J\ 

Y 

^^^ 

1 

\ 

\ 

'\ 

4 

1 

\ 

/ 

\ 

/ 

\ 

/ 

\ 

p, 

1 

\^' 

1 

\ 

x' 

0 

\ 

P* 

n 

/ 

X 

V 

J 

n 

Fig.  12. 


f: 


ints  are  the  solutions  of  the  equation,  O/*^  —  4  a;  +  3  =  0. 


296 


ELEMENTARY  ALGEBRA      [Ch.  XVIII,  §  234 


In  like  manner,  if  the  graph  of  any  equation  in  the  form 
y=ax'^+bx+c  is  plotted,  the  ir-coordinates  of  the  points  where 
the  graph  crosses  the  X-axis,  will  evidently  be  the  solutions  of 
aoc^  +  bx  +  c  =  0.  The  nature  and  approximate  values  of  the 
solutions  can  therefore  be  determined  from  the  graph. 


234.    (2)  When  P>4:  ae^  and  when  Vfi^  ^  ^  ac  i^  irra 
tional. 

Plot  the  graph  of  ^  =  x^—  ix  }-  2. 


Fig.  13. 


V, 

Y 

pA 

p^ 

Y 

P.      1 

\ 

i 

\ 

\ 

\ 

\ 

\ 

\ 

\ 

\ 

P. 

^8, 

\ 

^\ 

^8 

\ 

\ 

/ 

\ 

/ 

\ 

/ 

\ 

/ 

\ 

/ 

\ 

/ 

1\ 

Pj 

/ 

\ 

^. 

/ 

\ 

y 

y 

P-, 

a' 

0 

X 

\ 

/ 

r\ 

1\ 

/ 

\ 

i\ 

p. 

/ 

\ 

) 

x' 

0 

\ 

/ 

X 

P'. 

^5 

P. 

A 

n 

P, 

A 

Pa 

P, 

Ps 

A 

x= 

-2 

-1 

0 

1 

2 

3 

4 

5 

6 

y= 

14 

7 

2 

-1 

-2 

-1 

2 

7 

14 

Fig.  14. 

The  graph  is  constructed 
as  shown  in  Fig.  13,  and  is 
seen  to  cut  the  X-axis  at 
points  whose  o^-coordinates 
are  between  0  and  1,  and 
between  3  and  4.  By  the 
usual  method  of  solving  the 


Cii.  XVIII,  §§235,236]     QUADRATIC   EQUATIONS 


297 


equation  a^  —  4  a^  -4-  2  =  0,  the  roots  are  found  to  be  2  ±  V2,  or 
0.26795+  and  3.73205 4- .  These  must  therefore  be  the  exact 
values  of  x  where  the  graph  crosses  the  X-axis. 

235.    (3)  When  Jfl  =  4:ac,     . 

1.    Plot  the  graph  of  y  =  x^-'4:X+4i. 


A 

Pi 

Pz 

P, 

A 

P. 

P7 

P, 

A 

x= 

—2 

-1 

0 

1 

2 

3 

4 

5 

6 

y= 

IG 

9 

4 

1 

0 

1 

4 

9 

16 

A        Here  the  equation, 

a^-4a;  +  4  =  0, 

1^    has  equal  roots,  x=^2,  and 
the  graph.  Fig.  14,  touches 


the  X-axis  at  the  single  point  P5,  whose  coordinates  are  (2,  0). 

236.    (4)  When  P<4:ac. 
Plot  the  graph  of 


Px 

Pz 

Ps 

P4 

P5 

A 

i'r 

P. 

A 

x= 

-2 

-1 

0 

1 

2 

3 

4 

6 

6 

y= 

17 

10 

5 

2 

1 

2 

5 

10 

17 

If  2/=0,  and  the  resulting  equa- 
tion, i»2— 4  a;  +  5  =  0,  is  solved,  the 
roots  are  found  tobeic=2±V  — 1. 
Since  these  values  are  imaginary, 
they  cannot  represent  any  real  dis- 
tance. Hence  the  graph.  Fig.  15, 
does  not  cut  the  X-axis. 

The  graphs  of  the  equations 
which  have  been  plotted  have  the 
same  general  shape,  which  will  l)e 
found  to  be  the  same  for  all  equa- 
tions of  the  form  ?/  =  ax^  +  hx  -f  c. 
This  curve  is  called  the  parabola. 


Y 

Y 

'1 

\ 

\ 

\ 

\ 

\ 

^2 

P> 

1 

\ 

/ 

\ 

/ 

\ 

/ 

^, 

^1 

/ 

\ 

1 

\ 

\ 

A 

J% 

1 

\ 

/ 

x' 

0 

h 

X 

Fig.  15. 


298 


ELEMENTARY   ALGEBRA      [Cii.  XVIII,  §  2;J: 


GRAPHS   OF   EQUATIONS   CONTAINING  / 


)' 

y 

< 

T\ 

II 

^, 

^ 

1^ 

/ 

U 

•JJ 

fioS 

s. 

^ 

p 

^ 

j 

^1' 

r\ 

/ 

\ 

a' 

Pj 

0 

/;.. 

X 

\ 

\ 

\ 

// 

J 

^ 

<: 

7/ 

y 

K, 

s 

S^ 

I'l 

/ 

^/•,' 

< 

4 

li 

/>; 

^ 

^ 

'k 

L 

r' 

^5 

237.  1.  Plot  tne  graph 
of  .t2  +  ^2^36. 

Solving   a;2  +  ?/2^S6, 


■  a;- 


The  na- 
'^  is  sueh 


?/  =  ±  V3(3^ 
tare  of  V36- 
that  if  a;  takes  any  values 
less  than  —  6  or  greater 
than  o.  (^  oe comes  imagi- 
nary. Il  IS  necessary  to 
constnict  a  table  only  for 
values  of  x  between  —  6 
and  +6. 


Fig.  1(3. 


a;  = 

±6 

±5 

±4 

±3        ±2 

±1 

0 

2/  = 

0 

±vn 

±2V5 

±3V3'±4V2 

±V35 

C 

The  graph  is  constructed  as  shown  in  Eig.  16,  using  approxi- 
mations of  the  double  values  of  the  surd  values  of  ?/.  Points 
may  be  located  closer  together  by  taking  fractional  values  of 
a;,  as  1^,  If,  etc.     The  graph  is  seen  to  be  a  circle. 


:  2-2  —  6  r?T  +  9. 


EXERCISE  CXXII 

Plot  the  graphs  of  the  following  equations  : 

1.  ?/  =  2^2  — 7  a;  +  10.  6.   y 

2.  2/  =  .T2-3a:+5.  7.  a:2  +  ^2^25. 
Z,  y=:x^-1x-\-\.  8.  x^^y'^^X^. 
^,   y  =  x^—7  x+4:.  9.    y^  =  4:x. 

5,   y  =  x^-bx+6.  10.   x^+y^-x-S^O. 


CHAPTER   XIX 

SIMULTANEOUS   EQUATIONS   SOLVABLE   BY  QUADRATICS 
TWO   UNKNOWN   QUANTITIES 

238.  A  system  of  two  simultaneous  quadratic  equations 
involving  two  unknown  quantities  cannot,  in  general,  be 
solved  by  quadratics. 

r.     .  .  .  (X^-f=S,  (1) 

Solve  the  equations  :    \    ^      ^ 

Substituting  in  (2)  the  value  of  y  in  (1)  and  simplifying, 

x^  +  4ta^-lSx^-32x  +  67  =  0.  (3) 

Equation  (3)  cannot  be  solved  by  the  method  of  quad- 
ratics ;  and,  in  general,  the  solution  of  a  pair  of  quadratic 
equations,  chosen  at  random,  will  involve  the  solution  of 
an  equation  of  the  fourth  degree. 

There  are,  however,  certain  forms  of  simultaneous  equa- 
tions which  can  be  solved  by  means  of  quadratics. 

SIMULTANEOUS  EQUATIONS  SOLVABLE  BY  QUADRATICS 

239.  In  §  151  it  was  shown  that  the  coordinates  of  the 
point  of  intersection  of  two  lines  were  the  values  of  x  and 
y  in  the  solution  of  the  two  equations  which  the  lines 
represent,  since  the  coordinates  of  this  point  must  satisfy 
both  equations.  For  the  same  reason,  if  tlie  graphs  of 
two  quadratic  equations  or  a  simple  and  a  quadratic 
equation  are  plotted,  the  coordinates  of  the  points  of  in- 
tersection of  these  graphs  must    be  the  solutions  of   the 

pair  of  equations. 

299 


300 


ELEMENTAKY  ALGEBRA  [Ch.  XIX,  §  240 


240.    1.    Plot  the  m-aphs  of  the  system  :        „  '     ^  "^ 

""     ^  "^  U2  +  ^=5.   (2) 

By  §  145,  construct  the  graph,  AB,  of  x  -{-  y  =  3. 

By  §  237,  construct  the  graph  of  x^  -\- y  =  5. 


Pi 

P, 

P. 

P. 

^3 

P, 

Pr 

P. 

x  = 

0 

±1 

±V2 

±vs 

±2 

±V5 

±V6 

±V7 

y  = 

5 

4 

3 

2 

1 

0 

-1 

—  2 

Locate  suitable  points  and  draw  the  smooth  curve  PgPiP'^. 

The  intersections  of  the  graphs  AB  and  the  smooth  curve 

P'gPiPs,  Fig.  17,  will  be  points  whose  coordinates  are  solutions 

of  the  given  system. 

If,  in  place  of  a;  +  ?/=  3, 
the  graph  of  x-{-y  =  6  is 
plotted,  the  graph  will  be 
found  not  to  cut  the  parab- 
ola which  is  the  graph  of 
x^-]-y=5.  Corresponding 
to  this  non-intersection  of 
the  two  graphs  are  found 
imaginary  values  for  x 
and  y  when  the  equations 

\  \    are   solved 

simultaneously.  ^ 

If  the  graph  of  2  x-\-y—^ 
is  plotted,  the  graph  will  be  found  just  to  touch  the  parabola 
at  the  point  (1,  4).     Corresponding  to  this  fact,  if  the  equations 


\ 

r 

\ 

{', 

\ 

/ 

\ 

J\ 

/ 

k 

B^ 

\^ 

A 

i 

\ 

\] 

\. 

k 

N 

i\ 

X' 

n 

0 

A 

X 

// 

c 

/ 

\ 

h 

\  ' 

\ 

n 

V  ^ 

\ 

i 

\ 

/ 

\ 

/ 

\ 

1 

r 

r 

\ 

Fig.  17. 


l2x-\-y  =  ^ 
solution^  x=^l,  2/ =  4. 


^.9^1      _  A  ^^^  solved  simultaneously,  they  have  the  single 


€h.  XIX,  §  241]        SIMULTANEOUS    EQUATIONS 


801 


241.    1.   Plot  the  graphs  of  the  system : 

By  the  same  method 
nsed  in  the  preceding 
paragraph  the  graphs  of 
■the  two  equations  are 
plotted  as  shown  in  Fig. 

They  intersect  in  the 
four  points,  P^  P2,  P3, 
P4,  whose  coordinates  are 
found  by  measurement  to 
agree  with  the  sohitions 
of  the  two  equations, 

(4V2,  4),  (4V2,  -4), 

(-4V2,4),(-4V2,-4). 


;c2  +  2  2/2=64,(l) 
x^-f=16.     (2) 


^'                                       -y 

s                            y 

s                          z 

^                       z 

\^             ^^ 

M                J^ 

7     V                  74-v 

--l       A                      t      A 

jq    if                          0                           X 

T 

V      /                     V      / 

vp'          ^,y 

^^                    ^ 

y^-           --^     \ 

Z       X                   s 

/                                 s 

^                    -                 s 

31 

Fig.  18. 


The  graph  of  (1)  is  called  an  ellipse ;  of  (2),  an  hyper- 
bola. 

EXERCISE   CXXIII 

Plot  the  graphs  of  the  following  systems  and  deter- 
mine by  measurement  the  coordinates  of  their  points  oi 
intersection. 


2. 


3. 


xy  =  l. 
x  +  y  =  ^, 

'  a;2  +  ^2  ^  5^ 

xy==2, 

'  x^  —  y'^=^  16, 
.  2;  +  y  =  8. 


p2-, 
\xy  = 


^2=5, 
6. 


6. 


7. 


8. 


.xy 

(a^--y^=^2i, 
l3:i:2_20  7/2  =  55. 

( 4:X^  —  xy  =  6^ 
[S  xy  —  y^  =  6, 

cx^+y^  =  ll, 
Xx^  —  y^=  15. 


302  ELEMENTAEY   ALGEBRA  [Ch.  XIX,  §  242 

CASE   I 

242.    A  simple  equation  and  a  quadratic  equation. 

A  system  of  simultaneous  equations  in  which  one  equii- 
tion  is  simple  and  the  other  quadratic  can  always  bo 
solved  by  substituting  in  the  quadratic,  equation  the  vahu: 
of  one  of  the  unknowns  obtained  from  the  simple  equation. 

1.    Solve  the  equations  :    i    „      ^    „'      , 

Substituting  in  (2),  x  —  l  —  y  from  (1), 

(7-?/)2  +  2/=34,  .                   (3) 

simplifying  in  (?.),      ^y--Uy  +  15  =  0,  (4) 

factoring  in  (4),          (y  -  3)  (3  2/  -  5)  =  0,  (5) 

from  (5),  .  2/  =  3,  or  ?/  =  -.  (6) 

Substituting  values  of  y  from  (6)  in  (1), 

07  =  4,  or  a;  =  — .  (7) 

o 

The  given  equations  check  if  a;  =  4  and  y  =  3  be  substituted ; 

16  5 

and  the  given  equations  also  check  if  a?  =  —  and  ?/  =  -  be  sub- 

o  o 

stituted.     Such  values  of  the  unknowns  which,  taken  together^ 

satisfy  the  given  equations  are  called  dependent  values. 

Dependent  values  should  always  he  found  hy  substituting 
the  value  of  the  unknoum  first  found  in  the  simple  equation., 
and  never  in  the  quadratic  equation. 

It  is  to  be  noticed  that  the  given  equations  are  not  verified  by 
values  which  are  not  dependent. 


Cii.  XIX,  §  243]       SIMULTx\NEOUS   EQUATIONS 


303 


843.  The  use  of  the  double  signs,  ±,  read  "plus  or 
minus,"  and  T,  read  "minus  or  plus,"  taken  together  are 
to  be  interpreted  in  the  order  in  which  the  signs  are  read. 


Thus,        I 


x=±l,,  .  fX  =  +  l,  (X 

IS  equivalent  to  \  and   <; 


[y  =  ±^, 


i?y  =  +  ^^ 


is  equivalent  to  \  J  and 


/•  /^  IT    I 

Similarly  J  J  i~  -  . ^ 


-1. 

x  =  +  l, 
2/ =  -2. 


EXERCISE  CXXIV 

Solve  the  following  systems  of  equations 


1. 


'  xy  —  5  X  =:  1^ 


\2x^  +  xy=S. 


^  -  2/  =  4, 

3^-2_^^^2. 


5. 


6. 


7. 


2  a;  -  3  «/  =  2, 
3x2-2«/2=115. 

.  9  a;  +  7  z/  =  80. 
a^  +  22/  =  3, 


y 


£_+3j 

2:  +  2 


4a;+3y  =  l, 

^y  +  3 

X+1/ 


5.y  I  2y  +  3. 


0. 


10. 


1 


a;(J  —  a)      ^(a  +  b)      o?  —  b^ 

a      ^    2b 
Jj  +  ^b      x-y' 


=  01 


304 


ELEMENTARY  ALGEBRA      [Ch.  XIX,  §  244 


CASE  II 

244    When  one  of  two  simultaneous  quadratic  equations 
is  homogeneous. 

A  quadratic  equation  is  said  to  be  homogeneous  when  all  j 
of  the  terms  involved  are  of  the  second  degree  in  the] 
unknown  quantities. 

Thus,  x^  —  3xy  +  2y^  =  0  is  a  homogeneous  quadratic  equa- 
tion. 

x'^-xi/-2f  =  0,  (1)1 

(2)i 


1.    Solve  the  system :    ,    _ 


Dividing  (1)  by  y% 


factoring  (3), 


'x\^     fx^ 


yj    \y. 


2  =  0, 


^     2Y^  +  1]  =  0, 


(3): 

(4) 


from  (4),                                                        x=2y,  or  x=—y,  (6) 

substituting  x  =  2y  in  (2),             4:y^  +  y  =  5,  (6) 

solving  (6),                                                    y  =  l,  or  2/  =  -  i  (^) 

substituting  values  of  y  from  (7)  in  (5),    x  =  2,  or  x  =  —  f,  (8) 

substituting  x  =  —  y  in  (2),                 y^  -{-y  =  5,  (9) 

■l±V2l 


solving  (9), 


y- 


substituting  y  = "^     ^^  in  (5),  x  = 


2 
lTV2i 


-,     (10) 

(11) 


The  solutions  are : 


x  =  2,      rx=—^, 


X  =  ' 


1  1    7/  —  _  5 


2 
I+V2I 


I+V2I 
-I-V2I 


Dh.  XIX,  §  244]       SIMULTANEOUS   EQUATIONS 


305 


EXERCISE   CXXV 

Solve  the  following  systems  of  equations : 


3. 


4. 


5. 


6. 


7. 


.2:^2  +  3^2^11. 
'  2  2)2  __  3  ^^  +  y2  ^  0, 

'  a?  +  xy  =  0^ 
2x^-Sx-y  =  4:. 

(2x'^  +  xy-10y^=0, 

.x^+8xy  +  y  =  —  l. 

^x^-3xy+2y^=0, 
[xy  —  x  +  y  =  4:. 

Sx'^  +  x-y  =  29. 

x^^  xy  =  20  y\ 

x^  —  x  +  y  =  54. 


8. 


9. 


10. 


11. 


12. 


13. 


14. 


cfi  +  y'^  =  2xy^ 
2x^  —  xy  +  y  =  30. 


'  x^—  xy  —  2y'^: 


0, 


Bx'^  +  llxy  +  2y^  =  0, 

x^  —  xy  -\-  y  =  b. 

'ix^  +  1xy  ~  y'^=  0, 
x-2y  +  %y^=Z2. 

(lbx^-Uxy+\by^^O, 
,a;  +  2/-2«/2=_10. 

8  a;2  +  2  xy  -  3  «/2  =  Q, 
a^  +  a;  +  «/2  =  22. 

3  x2  +  8  a;?/  +  5  2/2  =  0, 
3x2  +  4a;^+^=_30. 


15. 


16. 


17. 


18. 


^2a;2  +  9a;«/  =  35«/2, 
,2x(x  +  «/)-ll?/  =  236. 

'  Qx'^=Wxy  +  35  «/2^ 

.  a;2  -  17  a;^/  -  180  y  =  -260. 

9a:2_39  2.^4.22z/2  =  0, 
I3a^-7a;  +  y  =  289. 

10a^  +  23a;«/  +  12y2=o, 
9a^  +  7a;«/  +  6jr  =  132. 


306  ELEMENTARY    ALGEBRA    [Ch.  XL\,  §§  245,  246 

CASE    III 

245.  When  each  of  two  simultaneous  quadratic  equa- 
tions is  homogeneous  only  in  the  unknowns  involved. 

A  sj^stem  of  two  simultaneous  quadratic  equations  which 
are  homogeneous  except  in  the  absolute  terms  may  be 
solved  as  in  Case  II,  by  combining  such  multiples  of  the 
two  equations  as  will  make  equal  the  absolute  terms. 

^2  +  ^y  ^  12,  (1) 

.xy-'ly^  =  \.  (2) 

Multiplying  (2)  by  12,  12  xy  -24.f-=  12,  (3) 

subtracting  (3)  from  (1),  x^  -  11  xij  +  24  t/-  =  0.  (4) 

Equation  (4)  may  be  solved  as  in  Case  II;  or  it  may  be 
solved  by  factoring. 

Factoring  (4),  (x-Sy)  (x  -  8  ?/)  =  0,  (5) 

from  (5).  x  =  3  y,  or  x  =  ^y,  (6) 

substituting  x  =  3y  and  x  =  Sy  in  (1),  and  solving  the  result- 

ing  equations,  ^  ^  ±  1^  2,  =  ±  ^V6, 

by  substitution  in  (6),  x  =  ±3,  x  =  ±  ^  V6. 


1.    Solve  the  system  :     \ 


,x  =  3, 
The  solutions  are  :  \ 

2/  =  l, 


x=—o, 


\^% 


^•=— |V6, 


2/=-l,  b=iV6,   l2/=-iV6, 


246.    An  alternative  method  for  solving  equations  of 
the  class  of  Case  III  is  called  the  vjc  method. 


1.    Solve  the  system  : 


|^  +  ^y  +  4/=6,  (1) 

Let  y  =  vx,  and  substitute  in  (1)  and  in  (2), 

x^  -^x'v  +  i:  x'v''  =  6,      (3)         3  0^2  +  8  xV  =  14,  (4; 

factoring,  a^{l+v +  4.v')  =  6,      (5)         x"" (3  +  S v')  =  U,  (6) 


Ch.  XIX,  §246]       SIMULTANEOUS   EQUATIONS  .             307 

equating  x^  in  (7)  and  (8),  ^_^^^_^^^,  =  g^^^,  (9) 

clearing  and  simplifying  in  (9),  4  w^  +  7  y  —  2  =  0,  (10) 

from  (10),                        v=\,ovv  =  -2,  (]  1) 
substituting  values  of  v  from  (11)  in  (7), 

6 


l  +  i  + 


¥f  T 


1  =  4,  (12) 


^-1_2  +  16~16'  ^^^^ 
extracting  square  roots  in  (12)  and  in  (13), 

(x==±2,   _  (14) 

ia;=±iVlO.  (15) 

When  v  =  \,  x=±2;  substituting  v=^  in  y  =  vx, 

2/  =  i(±2)=:±f  (16) 

When  v  =  —  2,  x  =  ±  ^VlO ;  substituting  v=—2  in  y=vx, 

.^  =  q:2VIO.  (17) 

The  solutions  are : 


07  =  2, 


(x  =  -2,     ra;  =  iVlO,  ra;  =  -|VlO, 


b  =  i,    U  =  -i   U  =  -fVio,    b  =  |Vio. 

The  values  of  x  inust  alivays  he  substituted  in  y  =  vx. 

Since  equations  of  the  type  of  Case  III  may  be  reduced  to 
a  quadratic  equation  homogeneous  in  all  its  terms,  and  since 
such  an  equation  may  always  be  expressed  as  a  quadratic  in 

-,  for  -  any  quantity  v  may  be^  substituted.     If  -  =  v,  x=vy\ 

y  y    y  y 


308 


ELEMENTARY   ALGEBRA  [Ch.  XIX,  §  247 


1. 


EXERCISE   CXXVI 

Solve  the  following  systems  of  equations : 

xy  =  ^- 


(x^  +  3xy  =  27, 


(x^  +  y 

\x^ 


3. 


4. 


'2  ==20, 
xy=  8. 

r  ^^  +  4  =  0, 

'  x^  —  xy  =  15, 
^x^-f  =  21. 

['22;2  +  :r7/  =  52, 
[2f-xy  =  S0. 


6. 


7. 


8. 


10. 


CASE  IV 


■f^ 


2a;2-2/2=17. 

(2x^-  31/2  =  60, 
.3x2- 4  0:3/  + ^2  =  64. 

'  6^2— 5a;?/  +  2  2/2=i2, 
32;2+22:y-32/2=-3. 
[2a;2_2x?/-2/2=3^ 
U2  +  3a;?/+y2=ii. 

3:r2_7^^_|_4^2__  _^ 
.2x^  +  xy-Sf  =  22. 


247.  When  two  simultaneous  quadratic  equations  are 
each  symmetric  with  respect  to  the  unknowns  involved. 

An  equation  is  said  to  be  symmetric  with  respect  to 
the  unknowns  involved  when  the  interchange  of  the 
unknowns  does  not  change  the  form  of   that  equation. 

Thus,  x^  +  xy  +  y'^  =  7,  and  xy  +  x  +  y  =  5,  are  symmetric 
quadratic  equations. 

A  solution  of  a  system  of  such  equations  may  always 
be  found  by  substituting  x  =  u  +  v,  and  j^  '=^u  —  v^  in  the 
given  equations. 

(X^  +  X7/-{-f=l,  (1) 

.xy  +  x  +  i/=:5.  (2) 

Let  x  =  u  +  Vf  and  let  y  =  u  — v. 


Solve  the  system  : 


Oil.  XIX,  §  247]       SIMULTANEOUS   EQUATIONS  309 

Substituting  x  =  u-\-v,  and  i/  =  ^  —  v,  in  (1)  and  in  (2), 

(21  +  vy  +  (u  +  v)(u-v)+(u-  vf  =  7,  (3) 

{u  +  v)  (u  —  v)  +  (it  -\-  v)  +  (u  ^  v)  =  5,  (4) 

simplifying  in  (3)  and  in  (4),         3u^  +  v'  =  7,                  ♦  (6) 

^f2  +  2u-v'  =  5,  (6) 

transposing  in  (6)  and  in  (6),                   v^  =  7  —  3u^,  (7) 

v^  =  u'  +  2u-5,  (8) 

equating  v^  in  (7)  and  in  (8),           7 ---Su^^u^ +  2u- 5,  (9) 

solving  (9),                                                   u  =  ^,  ov  u  =  —  2,  (10) 

Substituting  i^  =  |,  in  (5),       v  =  ±^, 

x  =  u  +  v  =  ^±^  =  2,  orl, 
And,  ?/  =  ^^-.'y  =  |-q:^  =  l,  or  2. 

Substituting  w  =  —  2,  in  (5),  'y  =  ±  V—  5, 

x  =  u  +  v  =  —  2±-\/—5, 
and,  2/  =  i^  — 2;  =  — 2q:  V— 5. 

The  solutions  are : 


x  =  2, 
J/  =  l, 


'a;  =  l,       fx  =  —  2  +  -V—5,       ra;  =  — 2  — V— 5, 
^2/  =  2,       \2/  =  _2-V^      {2/  =  -~2  +  V^=^. 


Two  simultaneous  quadratic  equations  which  are  sym- 
metric, except  in  respect  to  signs,  can  often  be  solved  by 
Case  IV. 

The  proof  that  equations  of  the  type  of  Case  IV  can  be 
solved  by  substituting  x  =  u  +  Vy  and  y  =  u—v,  is  beyond 
the  province  of  this  book. 


810 


ELEMENTARY   ALGEBRA 


[Cii.  XIX,  §  247 


EXERCISE   CXXVII 

Solving  the  following  systems  of  equations : 

^  +  ^^y  +  2/  =  29,  r  S(x^  +  ^^)  —  5xy  =  15, 


1^  +  ^ 


4. 


5. 


8. 


10 


11. 


12 


x^-^y^  —  x  —  y  =  22, 
^  +  y  +  ^y  =  —  1. 

'  ^y  +  ^(^  +  1 )  +  y  (2^  +  1)  =  24, 
^xy^^, 

xy-2x^-2y'^=^20, 
4,xy  +  x  +  y=:^'l^. 

3a;2  +  3/=8(:r  +  2/)-l, 
xy  —  X  —  y  =  \. 

x'^  +  y'^  +  xy  +  x  +  y  =  17, 

r^'2  +  /  -  3  a:^  +  2  :r  +  2  ^  =  9 . 

2:z:  +  2?/  +  ^^=16, 

'  ^^  +  2/^  +  ^  +  2/  =  62, 
.5:^^  +  4(^2  +  ^2)^328. 

r  2:2  +  2  :r2/  +  ^2  +  5  2;  +  5  ?/  =  84, 
.  0^  +  ^2  +  ^,  _^  ^  3^  32. 


l  +  i-  +  i  =  7, 

x^      xy      2/2 

I  :^:(2:  -y)J^y(x  +  y^=%+  xy. 

x^  +  y'^  +  x  +  y  =  cfi^ 


xy  +  x  +  y  = 


3a 


Ch.  XIX,  §248]       SIMULTANEOUS   EQUATIONS  311 

SPECIAL   DEVICES 

248.  Special  devices  may  be  employed  in  finding  solu- 
tions by  shorter  methods  for  some  of  the  systems  in  the 
preceding  cases,  as  well  as  for  certain  other  systems  whose 
equations  are  often  of  higher  degree  than  the  second. 

^  +  2/  =  3,  '  (1) 


1.    Solve  the  system :  ,    „        «      ^ 

Squaring  (1)  and  subtracting  from  (2), 

-2xy==  20,  (3) 

adding  (3)  and  (2),         o?-2xy  +  if  =  49,  (4) 

extracting  square  roots  in  (4),    x  —  y  =  ±l,  (5) 

-adding  (5)  and  (1),  a;  =  5,  or  —  2,  (6) 

subtracting  (1)  from  (5),  y  =  —  2,  or  5.  (7) 

r^r        1     .  f  x  =  5,         r  x  =  —  2, 

The  solutions  are :    .i  ^     \         ^ 

[y  =  -2,     l2/-=5. 


2.    Solve  the  system  : 


:^;3  +  ^3^1001,  (1) 

U+2/=ll.  (2) 

Dividing  (1)  by  (2),       x^-xy  +  y^  =  91,   .  (3) 
squaring  (2)  and  subtracting  from  (3), 

-Sxy^-SOy  (4) 
dividing  (4)  by  —  3  and  subtracting  from  (3), 

1^                                   x'^2xy  +  f  =  Sl,  (5) 

extracting  square  roots  in  (5),   x  —  y=  ±9,  (6) 
combining  (2)  and  (6),                        ic  =  10  or  1,  ?/  =  1  or  10.  (7) 

10,     (x  =  l, 


The  solutions  are :      \         ^        \ 

l2/=l.       U  =  10. 


812 


ELEMENTARY   ALGEBRA  [Ch.  XLY,  §  24l 


1. 


2. 


3. 


4. 


5. 


6. 


8. 


10. 


11. 


12. 


EXERCISE 

Solve  the  following  systems 

'  a;  +  ^  =  6, 
xy  =  5. 

x  +  y=20, 
xy  =  51. 

.  X2/  =  13. 

^2+  2/2  3^  34, 
a;^/  =  15. 

x'^  +  y'^=^  25, 
2  :z:?/  =  24. 

'  x  +  y^l2, 
^x^  +  y^=74. 

x-y  =  2, 

2^2.-2/2=20. 


:z;2  +  ^2^34^ 

x  +  y  =  8. 

X^  +  y^==74:, 
X  —  y  =2. 

xy  =  a, 
x^  +  y^  =  6. 

2:  +  1/  =  a, 
x^  —  y^  =  h, 

y^-\-xy=\b, 
x'^  +  xy  =  10. 


CXXVIII 

of  equations : 

p2  4- ^2^436^ 
[ri;  — 2/=  14. 
'  a;2  +  o;^  =  15, 


13. 


21. 


22. 


23. 


14. 


15. 


16. 


17. 


18. 


19. 


20. 


Vxy+y^ 


(  X' 


13^ 


xy  +  y'^ 
lx-^y=  —  2. 

x^  -i-Sxy^  28, 
xy  +  4:  y^  =  8. 

x  +  xy  +  y=29, 
x^  -i-xy  +  y^=  61. 

'  x^  +  xy  +  y^=  19, 
x^  —  xy  +  y^=  7. 
2  2:2  +  5  :ry  =  33^ 
2  y^  —  xy=  12. 

'  a;2  +  5  2:y  +  ?/2  =  43. 

.x^  +  5  xy  —  y^=  25. 


I2:  — 


y  =  2. 


,x  +  y  =  5. 


x  +  y: 

X     y 

-2  + -2  =  25. 


Ch.  XIX,  §  248]       SlMtTLTANEOtrS   EQUATIONS 


m 


24. 


25. 


26. 


27. 


28. 


29. 


30. 


31. 


32. 


33. 


^1   ^1 

-  +  -  =  a, 

X      y 

-^  +  -^  =  6. 

1+1  =  5, 

'  x^—  xy  =^  153, 

:r  +  ^  =  3, 

r  rz:^  +  :r^  =  10, 
Uz/  -  ^2  ^  -  3. 

'  ^  +  y  =  -  3, 

1+1  =  1 
X     y      b 

5  xy  =  84:  —  x^y^^ 
X  —  y  =  6, 

^  +  2/  -f-  3  =  0, 
U2  +  22/2=8. 

'1  +  1=2, 

x^-7/     2l' 
.2;  +  t/  =  7. 


34. 


35. 


36. 


38. 


39. 


40. 


y 

,  1      12 


X 


0^  +  -  =  !, 


a;z/  -  -  =  2, 

y 


xy 


-U  =  l. 


I  ha?  +  axy  =  a. 


a5  _  q 
xy 

[X     y 


41 
42 


0^—  y^  =9, 

x  +  y  +  Vx+~y  =  12, 
xy  =  20. 

a;^^  —  xy^  =  12, 
ofi  —  ^^z=  63. 


314  ELEMENTARY    ALGEBRA  [Cii.  XIX,  §  240 

THREE   OR   MORE   UNKNOWN   QUANTITIES 

249.*  Three  simultaneous  quadratic  equations  involving 
three  unknown  quantities  cannot  in  general  be  solved 
by  quadratic  equations.  The  solutions  of  certain  forms 
are  illustrated  in  the  following  examples. 

U2_^2  2/2_^2^5^  (1) 

1.    Solve  the  system  i    i  2x-^ y  +  z  =  6^  (2) 

[x  +  Ay-z  =  5.  (3) 

11  —  5  V 

Eliminating  z  in  (2)  and  (3),     x= — ^,  (4) 

o 

eliminating  x  in  (2)  and  (3),  z  =  -~  I'    }  (^) 

o 

substituting  x  and  z  from  (4)  and  (5)  in  (1), 

simplifying  and  solving  (6),     3/  =  1,  or  y  =  —  10.  (7) 

{x  =  2j 
Substituting  values  of  y  in  (4)  and  (5), 


2/  =  l, 

z  =  l, 


2/  =  - 10, 


2.    Solve  the  system  : 


'<2/+^)  =  -4,  (1) 

"K^  +  ^)  =  -10,  (2) 

i<^  +  2/)  =  -54.  (3) 

Dividing  the  sum  of  (1),  (2),  and  (3)  by  2, 

xy-\-yz-\-xz  =  —  34,  (4) 

30 

subtracting  (1)  from  (4),                         2/  = ?  (^) 

24 

subtracting  (2)  from  (4),                         a;=  --— ,  (6) 


Cii.  XIX,  §  240]       SIMULTANKOUS    EQUATIONS 
substituting  y  and  x  from  (5)  and  (6)  in  (4), 


-30- 

-24  =  -34, 

(J) 

solving  (7), 

z  =  ±6, 

(8) 

substituting  z  from  (8)  in  (G), 

x  =  Ti, 

(9) 

substituting  z  from  (8)  in  (5), 

y  =  T5. 

(10) 

EXERCISE   CXXIX* 


Solve  the  following  systems  of  equations : 


2. 


xy  =  —  42, 
a^2  =  48, 
yz  =  —  56. 

'  <y  +  ^)  =  8, 

y(.:?  +  2^)  =  18, 
y'^  +  z^  =  25, 

;32  +  :i;2  ^  20. 

xy  +  :r;2  +  2/^  =  3, 

{%x  +  2y  +  z  =  Q. 

a:  +  2/  =  15, 

^^  +  2  =  3, 

2:  +  ^2  =  8, 


7.    ^ 


9.    ^ 


10.    \ 


x^  +  rry  +  2/^  =  19, 

/  +  2/^  +  ^2  ^  37, 

U2  +  ^^+^^5=28. 

x^  +  xy  -{-z  =  2^ 
2^+2^  +  ^  =  3, 
rz:  — 2/  +  ^  =  0. 

^  +  2/2  +  ^^2  ^  21, 
xy  +  xz  +  yz  =^  14, 
x  +  2/  — ^  =  —  1. 

a^  +  2/  +  ^  =  4, 

xy  +  xz  +  yz  =  —  4^ 

X  —  y  +  z  =  8. 

xy  +  25?>fc  =  14, 
xz  +  yu=^  11, 
XU  +  yz  =  10, 
^  +  ^  +  ^  +  ^  =  10. 


S16 


ELEMENTABY  ALGEBRA  [Ch.X1X,§249 


1. 


REVIEW  EXERCISE   CXXX 

Solve  the  following  equations : 
x^—1  __  5r?;  —  1 


4.    a?-^8x-^ 


?.      ■\/4:X-S'-Vx+l  =  l 

5  2a;-3 


5.    a^x  —  2  &2  =  ab 


a^  +  1 
x{  I 


x^'2      2(0;- 2) 


7.    3  a; 


3 

6* 

2x  +  4: 


^     x^+1      a  +  h  .      c 

6.     = -| 


8. 


3a:  —  5 


a  —  c     X  —  a 


=  4 


X  c 

2a:-3 


«  +  S 


3  h(x  -  c) 


9.    Va:  +  3  +  Va:  +  6  —  Va:  +  11  =  0. 


10.    a;2  +  8a;  +  6Va;2  +  8a;-8-3  =  0. 


11. 


12. 


13. 


Solve  the  following  systems  of  equations : 

14.*J^a;  =  15, 


r4      7_8a 
X     y      xy 


■  a"" 


iofi  —  xy  =^5^ 


x^  +  xy^y==l, 
3a;  +  2i/-5  =  0. 


15. 


16. 


yz=20. 

ra:2  +  5^^  +  32/2==3, 
\  3  a:2  +  7  a:?/ +  4  /  =  5c 


17. 


a^2/  ==  - 1, 

4  a:2  +  (2  ?/  -  a;)  (2  2/  +  a;)  =  7. 


Ch.  XIX.  §  249]       SIMULTANEOUS  EQUATIONS 


317 


18. 


19. 


20. 


ay 


hx 


a  —  b 


X  +  a 

y 
h 


-b 


a 


.  -\/x  +  1  —  'y/x  —  1  =  -Vy. 
2  xy  +  4:  X  —  y  —  2  =  Q. 


21.    Construct  the  equation  whose  roots  are  — ^t ^nd 

1-V5  ^ 


22.  What  must  be  the  value  of  c  if  the  roots  shall  be 
equal  in  the  equation,  'ix^  —  2x  +  c=Q'l 

23.  Determine  the  values  of  k  if  the  roots  of  the  equation, 
kx^  +  2fe  —  3a;  +  2  =  0,  are  real  and  equal,  and  verify  the 
results. 

24.  Determine  without  solving  the  nature  of  the  roots 
of  2^2-3:^  +  5  =  0;  bx'^--Qx  +  l  =  0. 

25.  Find  the  values  of  k  in  order  that  the  equation, 
(x^  —  Zx+2')  +  k(x^  —  a;)  =  0,  may  have  equal  roots. 

26.  The     two     distinct     equations,    x'^  +  2  px  +  q  =  {)^ 
x^  -^  2  qx  -\-  p  =  0^  are   such   that   the   roots  of   the   first 
have    tiie    same    difference    as   the    roots  of   the   second 
Prove  that  either  j9  +  g  =  —  1,  ov  p  =  q. 


CHAPTER  XX 

PROBLEMS  INVOLVING  QUADRATIC  EQUATIONS 

250.  Since  the  two  roots  of  a  quadratic  equation  can 
be  rational,  irrational,  or  imaginary,  problems  solved  by 
means  of  such  equations  can  have  apparently  such  solutions. 
But  because  it  is  impossible  to  translate  all  the  restrictions 
expressed  or  implied  in  the  problem,  into  the  equations 
formed  from  the  conditions  of  the  problem,  solutions  must 
always  be  verified  by  substitution  in  the  problem  itself. 

EXAMPLES 

1.  One  of  the  two  factors  of  108  exceeds  the  other  by 
3.     Find  the  factors. 

Let  X  =  the  first  factor,  and  x  +  3  =  the  second  factor. 

By  the  conditions,  x(x  +  3)  =  108.  (1) 

Solving  (1),  cc  =  9  or  — 12;  whence  a;  +  3  =  12,  or  —  9. 

Hence  the  factors  of  108  are  9  and  12 ;  or  —  12  and  —  9. 

Each  of  the  above  solutions  satisfies  (1)  and  the  problem ;  but 
if  restrictions  were  imposed  that  both  factors  should  be  posi- 
tive, the  second  pair  would  be  rejected;  and  if  it  were  neces- 
sary that  factors  should  be  negative,  the  first  pair  would  be 
rejected. 

2.  A  company  of  76  men  and  boys  are  seated  in  chairs 
arranged  in  such  a  way  that  the  number  of  chairs  in  eacli 
row  is  3  more  than  twice  the  number  of  boys;  and  thai 

318 


Ch.  XX,  §  250]  QUADRATIC    EQUATIONS  319 

the  number  of  rows  is  4  less  than  the  number  of  boys. 
Find  the  number  of  boys. 

Let  X  =  the  number  of  boys. 

By  the  conditions,     (2  a;  +  3)  (a;  -  4)  =  76.  (1) 

Solving  (1),  a;  =  8,  or  —  |. 

The  restriction  implied  in  the  problem  is  that  the  solution 
shall  be  in  positive  integers,  since  it  is  absurd  to  speak  of  |  of 
a  boy.  Hence  the  root— f  must  be  rejected  as  a  solution  of 
the  problem. 

In  the  following  problems  if  possible  use  a  single  un- 
known, rather  than  several  unknowns. 

EXERCISE   CXXXI 

1.  The  product  of  a  number  and  its  half  is  18.  Find 
the  number. 

2.  The  product  of  the  third  and  seventh  parts  of  a 
number  is  21.     Find  the  number. 

3.  What  number  is  2^  times  its  reciprocal  ? 

4.  Find  a  number  the  sum  of  which  and  its  reciprocal 
is  2. 

5.  Find  a  number  the  sum  of  which  and  12  times  its 
reciprocal  is  8. 

6.  The  sum  of  the  squares  of  two  consecutive  integers 
is  145.     Find  the  numbers. 

7.  One  of  two  factors  of  a  number  exceeds  the  oth3r 
by  2.  If  the  product  of  the  factors  is  80,  find  tlie 
numbers. 


320  ELEMENTARY   ALGEBRA  [Ch.  XX,  §  250 

8.  The  product  of  two  factors  of  a  number   is    18|. 
Find  these  factors  if  one  factor  exceeds  the  other  by  5. 

9.  The  sum  of  two  numbers  is  9,  and  their  product  is 
18.     Find  the  numbers. 

10.  The  sum  of  two  numbers  is  7,  and  the  sum  of  their 
squares  is  29o     Find  the  numbers. 

11.  The  difference  of  two  numbers  is  7,  and  their  prod 
uct  is  120.     Find  the  numberso 

12.  The  difference  of  two  numbers  is  4,  and  the  dif- 
ference of  their  squares  is  72.     Find  the  numbers. 

13.  The  sum  of  two  numbers  is  8,  and  the  sum  of  their 
cubes  is  152.     Find  the  numbers. 

14.  Find  two  numbers  such  that  the  sum  of  the  num- 
bers and  the  difference  of  their  squares  is  11. 

15.  Find  two  numbers  such  that  tlieir  sum  is  15,  and 
their  product  is  36. 

16.  If  the  length  and  breadth  of  a  rectangle  are  each 
increased  by  4  feet,  the  area  is  increased  by  100  square 
feet ;  but  if  the  length  and  breadth  are  each  diminished 
by  1  foot,  the  area  is  88  square  feet.     Find  the  dimensions. 

17.  A  rectangle  whose  area  is  160  square  inches  is 
surrounded  by  a  border  2  inches  wide.  The  border 
contains  120  square  inches.  Find  the  dimensions  of  the 
rectangle. 

18.  The  diagonal  of  a  rectangle  is  50  feet,  and  the 
perimeter  is  140  feet.     Find  the  area, 

19.  Find  the  length  of  a  rectangle  whose  area  is  11  Gl 
square  feet,  if  the  sum  of  its  length  and  breadth  is  70  feet. 


Ch.  XX,  §250]  QUADRATIC   EQUATIONS  321 

20.  A  number  of  men  each  subscribed  a  certain  amount 
to  take  up  a  deficit  of  $100;  but  5  men  failed  to  pay  and 
thus  increased  the  share  of  the  others  by  |1  each.  Find 
the  share  of  each. 

21.  It  took  as  many  days  to  do  a  piece  of  work  as  there 
were  men ;  but  if  there  had  been  4  more  men,  these  men 
could  have  done  the  work  in  9  days.  Find  the  number 
of  men. 

22.  Divide  10  into  two  such  parts  that  their  product 
shall  be  12  times  their  difference. 

23.  A  number  exceeds  a  second  number  by  4.  Find 
these  numbers  if  the  sum  of  their  reciprocals  is  -^^, 

24.  In  a  number  of  two  digits  the  units'  digit  exceeds 
the  tens'  digit  by  4,  and  the  product  of  the  number  and 
the  tens'  digit  is  192.     Find  the  number. 

25.  A  can  do  a  piece  of  work  in  3  more  days  than  B; 
and  both  can  do  tlie  work  in  5^  days.  How  long  will  it 
take  each  alone? 

26.  Divide  10  into  two  such  parts  that  the  quotient  of 
10  and  the  greater  part  equals  the  quotient  of  the  greater 
and  less  part. 

27.  The  quotient  of  a  number  of  two  digits,  divided  by 
the  sum  of  the  digits,  is  6  ;  and  if  the  sum  of  the  squares 
of  the  digits  be  subtracted  from  the  number,  the  remain 
der  is  13,     Find  the  number. 

28.  A  sold  goods  for  $56,  and  gained  as  many  per  cent 
is  the  goods  cost.     How  much  did  the  goods  cost  ? 

29.  A  number  exceeds  a  second  number  by  5 ;  the  differ- 
ence of  their  cubes  is  665.     Find  the  numbers. 


322  ELEMENTARY   ALGEBRA  [Ch.  XX,  §  250 

30.  Separate  250  into  two  such  numbers  that  the  sum 
of  their  square  roots  shall  be  22. 

31.  If  A  had  sold  7  books  less  for  $42,  he  would  have 
received  $1  a  book  more.     Find  the  price  of  each  book. 

32.  A  sold  a  number  of  yards  of  cloth  for  $40.  Had 
the  price  of  a  yard  been  50  cents  less  he  could  have  sold 
4  more  yards  for  the  same  money.  Find  the  price  per 
yard. 

33.  A  bought  two  pieces  of  cloth,  which  together  meas- 
ured 36  yards.  Each  piece  cost  as  many  dollars  per  yard  as 
there  were  yards  in  the  piece,  and  the  cost  of  the  first  was  4 
times  the  cost  of  the  second  piece.  Find  the  number  of 
yards  in  each  piece. 

34.  A  can  row  in  still  water  1|  miles  an  hour  faster 
than  the  current.  It  takes  him  8  hours  to  make  a  round 
trip  of  18  miles.     Find  the  rate  of  the  current, 

35.  A  tap  A  can  fill  a  cistern  in  9  minutes  less  than 
a  second  tap  B  can  empty  it.  If  A  and  B  are  running, 
it  takes  3  hours  to  fill  the  cistern.  How  long  will  it  take 
B  alone  to  empty  it  ? 

36.  In  a  number  of  two  digits  the  tens'  digit  is  double 
the  units'  digit ;  and  if  the  number  be  multiplied  by  the 
sum  of  the  digits,  the  product  is  567.     Find  the  number. 

37.  Find  two  numbers  whose  difference  multiplied  by 
the  greater  produces  35,  and  whose  sum  multiplied  by 
the  less  produces  18. 

38.  What  is  the  price  of  eggs  when  10  more  for  f  1 
lowers  the  price  4  cents  per  dozen  ? 


Ch.  XX,  §250]  QUADRATIC   EQUATIONS  323 

39.  A  sum  of  money  at  simple  interest  for  1  year 
amounted  to  $20,800;  if  the  rate  were  1^  less,  the 
amount  would  be  $200  less.  Find  the  principal  and 
the  rate  per  cent. 

40.  A  party  of  friends  went  on  a  pleasure  excursion, 
the  expense  of  which  they  share  equally.  If  the  number 
of  the  party  had  been  decreased  by  7,  and  if  the  total 
expenses  had  been  $150,  the  assessment  for  each  person 
would  have  been  $1  more  than  it  was;  but  if  the  num- 
ber of  the  party  had  been  increased  by  8,  and  if  the  total 
expense  had  been  $160,  the  assessment  for  each  person 
would  have  been  $1  less  than  it  was.  Find  the  number 
of  the  party,  and  the  assessment  for  each  person. 

41.  A  and  B  had  a  money  box  containing  $210,  from 
which  each  drew  a  certain  sum  daily  —  this  sum  being 
fixed  for  each,  but  different  for  the  two.  After  6  weeks, 
tlie  box  was  empty.  Find  the  sum  which  eg-ch  drew  daily 
from  the  box,  knowing  that  A  alone  would  have  emptied 
it  5  weeks  earlier  than  B  alone. 

42.  On  a  certain  road  the  telegraph  poles  are  placed 
at  equal  intervals,  and  their  number  per  mile  is  such  that 
if  that  number  were  less  by  1,  each  interval  between  two 
poles  would  be  increased  by  2\^  yards.  Find  the  number 
of  poles,  and  the  number  of  intervals  in  a  mile. 

43.  A  broker  sells  certain  railroad  shares  for  $3240.  A 
few  days  later,  the  price  having  fallen  $9  per  share,  he 
buys,  for  the  same  sum,  5  more  shares  than  he  had  sold. 
Find  the  price  and  the  number  of  shares  transferred  on 
each  day. 


CHAPTER  XXI 

RATIO,  PROPORTION,  VARIATION 
KATIO 

251.  The  ratio  of  one  number  to  another  number  is  the 

quotient    obtained   by  dividing   the   first   by  the    second 
number.     The  quotient  shows  how  the  numbers  compare. 
Thus,  the  ratio  of  5  to  T  is  indicated :  5  ^  7,  -f^,  5  :  7. 

The  ratio  of  one  quantity  to  another  quantity  of  the  same 
kind  is  the  ratio  of  the  numerical  values  of  the  quantities. 

Thus,  the  ratio  of  a  dollars  to  h  dollars  is  -. 

h 

The  terms  of  a  ratio  are  the  terms  of  the  fraction  indi- 
cating the  ratio ;  the  numerator  is  called  the  antecedent, 
and  the  denominator  the  consequent  of  the  ratio. 

Thus,  a  and  h  are  the  terms,  a  is  the  antecedent,  and  h 
the  consequent  of  the  ratio  ^. 

There  is  no  ratio  of  one  quantity  to  another  of  a  different 
kind^  since  it  is  impossible  to  compare  such  quantities. 

Thus,  no  ratio  exists  between  a  inches  and  h  pounds. 

252.  If  the  ratio  of  two  quantities  can  be  expressed  as 
a  rational  number,  they  are  said  to  be  commensurable ;  if 
the  ratio  of  two  quantities  is  an  irrational  number,  they 
are  said  to  be  incommensurable o 

32^ 


Ch.  XXI,  §  253]  HATIO  S25 

Thus,   when    -  =  -,   a  and   h    are    commensurable;    when 
h      4 

2:=  V2,  a  and  h  are  incommensurable. 

The  ratio  of  two  commensurable  quantities  is  called  a 
commensurable  ratio;  the  ratio  of  two  incommensurable 
quantities  is  called  an  incommensurable  ratio. 

Thus,  when  -  =  5  and  when  -  =  V3,  5  and  V3  are  respec- 
b  b 

tively  commensurable  and  incommensurable  ratios. 

253.*  An  incommensurable  ratio  can  always  he  expressed 
as  a  commensurable  ratio  whose  value  differs  from  the  in- 
commensurable  ratio  by  less  than  any  assigned  quantity^ 
however  small. 

If  a  is  a  diagonal  of  a  square  of  which  J  is  a  side,  -  =  V2. 

b 

In  §  185  it  was  shown  that  V2  may  be  determined  to  any 

equired  degree  of  accuracy. 

In  general,  let  a  and  b  be  any  two   incommensurable 

quantities.     Let  p  be  contained  in  b  integrally  (say)  m 

imes,  and  let  p  be  contained  in  a  more  than  (say)  n  times, 

ind  less  than  n  +  1  times.     That  is,  let, 


mp  =  6, 

(1) 

np<a^ 

(2) 

a<(n  +  l)p. 

(3) 

Dividing  (2)  by  (1),  ^<^,  or  ^>^, 
m.     0          0     m 

(4) 

fividing  (3)  by  (1),      ?<-  +  -•  (5) 

'  0     m     m 


326  ELEMENTARY  ALGEBRA  [Cii.  XXI,  §  25^ 

Since  from  (4),  7  >— ,  and  from  (5),  -< — | — ,   -  dif 
b      m  0      m      m     0 

fers  from  —  by  less  than  — ;  or, <  —  (6) 

m  m  0      m     m 

Since  it  is  always  true  that  mp  =  S,  by  taking  p  smaller 

and  smaller,  m  will  increase :  hence  —  will  decrease  and 

m 

may  be  made  less  than  any  assigned  quantity.     Therefore 

-  can  be  made  to  differ  from  the  commensurable  ratio  — . 
0  m 

by  less  than  any  assigned  quantity,  however  small. 

Note.     If  p  is  very  small,  -  is  nearly  equal  to  — ;  but  -  :^  — . 

b  m  b       m 


254.  The  reciprocal  of  a  given  ratio  is  called  an  inverse 
ratio. 

Thus,  f  is  the  inverse  ratio  of  -J. 

A  ratio  of  equality  is  one  in  which  the  antecedent  and 
consequent  are  equal ;  a  ratio  of  greater  inequality  is  one 
in  which  the  antecedent  is  greater  than  the  consequent ; 
a  ratio  of  less  inequality  is  one  in  which  the  antecedent  is 
less  than  the  consequent. 

Thus,  f,  I,  I,  are  respectively  ratios  of  equality,  greater 
inequality,  and  less  inequality. 

The  ratio  found  by  squaring  the  terms  of  a  given  ratio 
is  called  a  duplicate  ratio ;  the  ratio  found  by  cubing  the 
terms  of  a  given  ratio  is  called  a  triplicate  ratio. 

2  3 

Thus,  — ^  and  —^  are  respectively  the  duplicate  and  the  trip- 
licate ratios  of  -  • 
h 


Oh.  XXI,  §  254]  RATIO  327 

EXERCISE   CXXXII 

1.  Express  the  ratio  of  5  to  7;  4|  to  12;  6  to  1 ;  3j\ 

to  71. 

2.  Express  the  ratio  of  a  cents  to  h  cents ;  m  inches  to 
n  inches ;  c  dollars  to  a  dollars ;  m^  feet  to  i\^  inches. 

3.  Determine  which  of  the  following  ratios  are  com- 
mensurable : 

2     m     6V2     6V2     _5i      (Ts/h 
3'    n       V2'     V3'    2^\'    ^Vi' 

4.  Determine  which  of  the  following  ratios  are  incom- 
mensurable : 


m 


12  V5       1        11        16       V9     V:J  +  1 


- '       / — 


n       V5       V8     V7     V16     V4     V3-1 

5.  Find  both  the  duplicate  and  triplicate  ratios  of: 

c     a/3     6     V3     ]V3     V2     Vm 

1:  T"'  7'  v5'  ^'  Jn   W 

6.  Determine  which  of  the  following  ratios  are  those  of 
jreater  inequality  and  which  are  those  of  less  inequality ; 

2    6     9    a     V^    V8     V5  +  4         4 
4'  7'   8'   6'    d'     2  '      VM    '   2  +  V5" 

7.  Prove  that  a  ratio  of  greater  inequality  is  diminished 
f  the  same  positive  quantity  is  added  to  both  terms. 

8.  Prove  that  a  ratio  of  greater  inequality  is  increased 
f  the  same  positive  quantity  is  subtracted  from  both 
arms. 

9.  Prove  that  a  ratio  of  less  inequality  is  increased  if 
he  same  positive  quantity  is  added  to  both  terms. 


S28  Elementary  algebea   [Ch.  xxi,§§255,25e 

PROPORTION 

255.  A  proportion  is  an  equation  whose  members  are 
ratios.  A  proportion  may  be  expressed  thus:  7  =  ^1 
a  :  b  =  e  :  d^  a  :  b  :  :  c  :  d. 

The  terms  of  the  equal  ratios  forming  a  proportion  are 
called  the  terms  of  the  proportion.  The  antecedents  and 
consequents  of  the  ratios  are  called  the  antecedents  and ; 
consequents  of  the  proportion.  The  first  and  fourth  terms 
of  a  proportion  are  called  the  extremes  and  the  second 
and  third  terms  are  called  the  means.  The  terms  of  a 
proportion  are  said  to  be  proportional.  The  fourth  term 
of  a  proportion  is  called  a  fourth  proportional.  When  the 
second  and  third  terms  of  a  proportion  are  identical,  it 
is  called  a  mean  proportional,  and  the  consequent  of  the 
second  ratio  is  called  a  third  proportional. 

Thus,  in  the  proportion,  -  =  -,  a,b,  c,  and  x  are  proportional, 

b      X 

a;  is  a  fourth  proportional;   in  the  proportion,   -  =  -,  a?  is   a 

third  proportional,  and  6  is  a  mean  proportional. 

A  continued  proportion  is  a  series  of  equal  ratios  in 
which  the  consequent  of  each  ratio  is  the  antecedent  of 
the  next  ratio. 

Thus,  -  =  -  =  -  is  a  continued  proportion. 
bed 

256.*  If  two  incommensurable  ratios^  -  and  — ,  are  so  related 

b  d 

71  n      a      n       A-         7 

to  the   commensurable  ratio    — ,  that   —  <-< — I ,   ivhen 

w.  m     b      m      m 

-^<£.<  — H ,  however  much  n  and  m  are  increased^  then 

m     d     m      m 

a  _c 

b~d' 


Cii.  XXI,  §  2o7]  PROPORTION  329 

If  7^-,  since  both  -  and  —  lie  between  —  and  — | — , 
h      d  0  d  m  m     m 

their  difference  must  be  some  quantity  less  than  — .     But, 

^  m 

since  m  can  be  made  to  increase,  —  can  be  made  less  than 

m 

any  assigned  quantity:  hence  - — -  can  be  made  as  small 

0      d 

as  is  required;  a  fact  which  is  true  only  when  7  =  -. 

0      d 

Two  incommensurable  ratios  are  therefore  equal  under 
the  conditions  named  above,  and  hence  may  form  a 
proportion. 

PRINCIPLES   OF  PROPORTION 

257.  I.  In  any  proportion  the  .product  of  the  means 
equals  the  product  of  the  extremes. 

If  2= J,  (1) 

multiplying  (1)  by  ScZ,  ad  =  be.  (2) 

II.  If  two  products  are  each  composed  of  two  factors, 
thei^e  factors  form  a  proportion  in  which  the  factors  of  either 
product  can  be  made  the  means^  and  the  other  two  factors 
the  extremes. 

0) 

(2) 


If 

ad^ 

=  bc^ 

dividing  (1)  by 

bd, 

a 

b" 

_  c 
''d' 

Similarly, 

h 
d 

a 
c 

a  _ 
e 

h 

etc 

330  ELEMENTARY    ALGEBUA  [Cii.  XXT,  §  257 

III.    The  products  of  corresponding  terms  of  two  or  more 
proportions  are  in  p)ro port  ion. 

If  ^^  =  |,  0) 

0     a 

and  II  _=  ~,  i^Z) 

n      s 

1       A    '        o  a     m      c     r  ^o\ 

by  Axiom  3,  t  *  —  =  "^  *  ~'  \^) 

^  b     n      a     s 


am      cr 


or,  rewntmg  (3),  -  =  -■  (4 


IV.  T/i6    quotients   of  the    corresponding   terms  of  two 
proportions  are  in  proportion. 

If  •      f=!'  O) 

and  if  ^  =  ^,  (2) 

1       A    •        ^  a      m      c      r  ^q\ 

by  Axiom  4,  7  -^  —  =  -  —  -9  v^>' 

•^  b      n      d      s 

or,  simplifying  in  (3),         ^  =  £*  ("^^ 

V.  If  four  quantities,  a,  b,  c,  d,  are  in  proportion,  they 

7      .  •         ^7   J.  '     b      d 

are  m  proportion  by  inversion  ;  that  is,  -  =  — 

If  ^=S'  (^> 

b      d 


by  I9  ad  =  J^,  (2) 

d      b  b      d 

-  =  -,  or      : 

c      a         a 


dividing  (2)  by  a^,        _  =  -,  or     =r  ^-  \  3) 


Cn.  XXT,  §  257]  PROPORTION  831 

VI.    If  four  quantities  of  the  same  Jcind^  a,  J,  ^,  c?,  are  in 
proportion^  they  are  in  proportion  by  alternation ;  that  is^ 


(1) 

(2) 
(3) 


a  h 
0      d 

If 

a  e 
b~d' 

by  I, 

ad  =  be^ 

dividing  (2)  by  cd, 

a  _b 

c 


Note.     ^2JH£^  ^  10  pounds  ^^^^^^  ^^  written  by  alternation, 
3  inches       6  pounds 

5  inches     •     •  -u i 

since  IS  impossible. 

10  pounds 


VII.    If  four  quantities^  a,  5,  c^  6?,  are  in  proportion^  they 

\  in  proportic 

a  +  b      c  -\-  d 


are  in  proportion  by  composition  ;  that  is^      ";      =  -~ — 
,              ,  b  d 

a  4-  0       (J  4-  d 
or 


a  c 

adding  1  to  each  member  of  (1), 

2  +  1=1  +  1.  (2) 

•x*       /'Ci'\  a  +  b      c  A- d  ^o\ 

or,  rewriting  (2),  —^  =  -=^.  (3) 

0  a 

Similarly,  (1),  written  first  by  V,  and  then  by  composi- 

..        .     a  +  b     c  +  d 
tion,  IS  — ' —  =  — = — r 


332  ELEMKNTARY   ALGEBRA  [Cii.  XXI,  §  257 

Vlil.   If  four  quantities,  «,  6,  c,  d^  are  in  proportion^  they 
I      J-  '^'^       J.T  J.    '      a  —  h      c  —  d 


(1) 

(2) 
(3) 
(4) 

(5) 
(6) 

(7) 


IX.   If  four  quantities^  a^  5,  c,  c?,  are  in  proportion^  they 
are   in  proportion   by   composition   and  division;    that    is^ 
o^-t-  h  _  c  +  d 
a  —  b      c  —  d 

If  |=|,  (1) 

0     a 


a—b      c—d 

b 

d 

a             c 

If 

a      0 
b'^d' 

subtracting  1  from  each  member  of  (1), 

f->"j-^' 

Or,  rewriting  (2), 

a—b      c—d 
b            d    ' 

\ 

writing  (1)  by  V, 

b_d 
a     e 

subtracting  1  from  each  member  of  (4), 

^-1  =  ^-1, 
a             c 

or,  rewriting  (5), 

b  —  a     d  —  c 

5 

a             c 

multiplying  (6)  by  • 

-1, 

a—b      e—d 

Ch.  XXI,  §  257]  PROPORTION  338 

writing  (1)  by  VII,     ^  =  ^,  (2) 

writing  (1)  by  VIII,    ^  =  "-^,  (3) 

;.;IV,  a  +  h^c  +  d  .^. 

a—bc—d 

X.  Like  powers^  or  like  roots  of  four  quantities^  a,  I\  (?,  d^ 
which  are  in  proportion^  are  in  proportion;  or  —-  =  — 

If  2  =  1  (1) 

raising  each  member  of  (1)  to  the  nth.  power,  whetht^r  n  is 
integral  or  fractional, 

l'^  ""  d"" 

XI.  In  a  series  of  equal  ratios  the  sum  of  the  antecedents 
is  to  the  sum  of  the  consequents  as  any  antecedent  is  to  its 
own  consequent. 

If  cb __  c ^m __x  ^-l^^ 

II  —  —  — — ,  {^y 

0      d      n      y 

1  i.  a  c  m  X  ,o\ 

let  -  =  r,  -  =  r,   -  =  r,  -  =  r,  (2) 

0  d  n  y 

clearing  of  fractions  in  (2),  a  =  hr^  c  =  dr^  m^nr^  x=^yr^   (3) 

by  Ax.  1,      a  +  c  +  m  +  x  =  (J)  +  d  +  n  +  y^r,  (4) 

dividing  each  member  of  (4)  by  {I  +  d  •\-n  +  y), 

a-^  c-\-  m-^-x  _    _a  _c__^m  _x  ^r, 

h  -f  d-\-n-{-y  b      ^     n     y 


334  ELEMENTARY   ALGEBRA  [Ch.  XXI,  §  257 


EXAMPLES 

1.    Solve  for 

V2a;+8-V2a:-5      1 
Xs =  -. 

(1^ 

V2a;  +  3  +  V2a;-6      2 

V      J 

By  IX, 

2V2a;  +  3          3 
-2V2a;-5      -i' 

(■/■ 

simplifying  (2), 

V2a;  +  3  =  3V2a;-5, 

(3J 

solving  (3), 

x  =  3. 

(4) 

2     If*-^ 
"         h~d' 

prove  that  ^^^_^^^^-^^. 

(1) 

Byx, 

0?      (? 

(2) 

since  ^  =  ^  =  1, 

il  =  i, 
P     Q 

(3) 

by  III, 

pa?  _  qc' 
pb"""  qif 

(i) 

by  VI, 

pa?  _  pb^ 
qc"  ~  qcP' 

(5) 

by  VIII, 

pa^  —  qe^  _  pb^  —  qd? 
q(?                qrP      ' 

(6) 

by  VI, 

pa'  —q(?_ q<?  _<?  _ a?^ 
pb^-qcP     qdP     cP      6« 

(7) 

An  alternative  method  for  this  example  is : 

by  I, 

a-lrp  —  bh-q  =  aWp  —  a?cPq, 

(2) 

simplifying  (2), 

h''(?  =  aH\ 

(3) 

byX, 

be  =  ad, 

(4) 

The  first  method  is  preferable. 


Cii.  XXI,  §  257]  PROPORTION  336 

EXERCISE    CXXXIII 

1.  Find  a  fourth  proportioiuil  to  462,  77,  and  90. 

2.  Find  a  third  proportional  to  35  and  91. 

3.  Find  a  mean  proportional  to  2  +  V3  and  2  — -  V3. 

4.  bolve  tor  X  : -^ = . 


5.  Solve  for  x : 

6.  Solve  for  a 
^^B.   Solve  for  x : 


'Sx-1                Sx-16     ' 
5  ^^  __  4  ^  +  1 0  _  5  ;7.^  -  4  .T  -  2 

Va  +  5  +  V5  —  ^  _  o 

Va  +  5  —  V5  —  a 

Va  —  hx-\-Vc  —  mx      -Va—hx—^e- 

■  mx 

Va  —  hx+  ^nx  —  d      Va  —bx  —  Vnx  —  d 


If  7  =  -,  prove  that : 
0      d 


8.    —^^-^ —.  10. 


a2 

a2- 

-62 

C2 

c2- 

-(f2- 

a2 

+  C2_ 

_  ac 

ah-^cd 

a2 
"a2 

+  ^2 

ah  —  cd 

-c2- 

a?+ah_ 

J2 

-2a6 

^     Ifi^d^     hd  ^^*    6^2+^^     d:^-^2ed 


12. 


13. 


14. 


15. 


a  +  h__a  —  b__a_^  h 
c+  d      c—  d      c      d 

a+h  +  c+  d  __  a—  b  -^  e—  d 
a+  b  —  c—  d      a—b—c+d 


-\a?  -\-  (^  __  c 

d^  —  ab  +  b^     6'2  —  cd  +  d^ 


336  ELEMENTARY   ALGEBRA  [Ch.  XXI,  §  257 


If  -  =  -  =  —,  prove  that  : 
h      d     f 

a  +  c  +  e  _a  c?  -\-  (?  -{-  p?  _  ace 

'    b  +  d  +  f^b  '    h^^d^^fbdf 

Ivvo  -{-kc  +  le  __a  mc?  +  nc^  +  pe^  __  clc^^ 

•    j^j^j^M  +  lf^V  '    ynb^  +  nd^+pP"  bd 

JO  a      b      c  ii4.<35  +  S      b  +  c 

20.    If -  =  -  =  -,  prove  that -—^ —  = -. 

b      c      d  0  +  c      c  +  d 

^^  X       V 
21 


.   If  ^  =  f  =  g  =  yfc,  prove  that  ^^^+9^^+^^^^ 

22.  If  -  =  -,  if  a:  is  a  third  proportional  to  a  and  6,  and 

b      d  7 

if  2/  is  a  third  proportional  to  c  and  cZ,  prove  that  —  =  -^• 

y     d 

23.  What  is  the  ratio  of  the  mean  proportional  between 
a  and  J,  to  the  mean  proportional  between  c  and  d  ? 

24.  Two  numbers  are  as  3:4,  and  if  7  be  subtracted 
from  each,  the  remainders  are  as  2  :  3.     Find  the  numbers. 

25.  What  two  numbers  whose  difference  is  d  are  to 
each  other  as  a  :  J  ? 

26.  Two  numbers  x  and  y  (the  first  being  negative)  are 
in  the  ratio  8  to  —  9  :  if  16  be  subtracted  from  each  one, 
the  resulting  numbers  are  in  the  ratio  —  9  to  8  ;  find  the 
numbers. 

27.  If      ^     =     ^     =  — — ,  prove  that  x-{  y  +  z=^^  0. 

a—  b      b  —  e      c—  a 

28.  If  — - —  =  — y- —  =  — - —  =z  1,  provf^  that 

a^  —be      S^  —  ca      (?  —  ab 

{a  +  b  +  c^{x  +  y  +  z)=^a^  +  b^  +  c^-  3  (fhc. 


Oh.  XXI,  §§  258, 259]  VARIATION  33'} 

VARIATION 

258.  A  quantity  whose  value  is  dependent  upon  the 
value  of  another  quantity  is  called  a  function  of  that 
quantity. 

Thus,  \i  y  =  2  Q(?,  y  \^  called  a  function  of  x. 

A  function  of  x  is  indicated  in  any  of  the  following 
ways:  F(x),  f(x),  ^(x),  etc. 

When  the  value  of  a  quantity  is  always  the  same  in  a 
particular  investigation,  the  quantity  is  called  a  constant. 

Thus,  X  is  a  constant  whose  value  is  2,  in  2  a;  +  5  =  7  +  x. 

When  the  value  of  a  quantity  changes  in  a  particular 
investigation,  the  quantity  is  called  a  variable. 

Thus,  in  the  expression  x^  + 1,  x  is  a  variable,  since  it  may 
take  any  value. 

The  theory  of  the  dependence  of  a  quantity  upon 
another  quantity  is  called  variation,  or  functionality. 
Only  the  simplest  forms  of  variation  are  discussed  in  this 
chapter. 

The  symbol  oc,  called  the  symbol  of  variation,  is  used  to 
indicate  variation. 

Thus,  a;  oc  2/  is  read  "  x  varies  as  yJ^ 

KINDS   OF  VARIATION 

259.  1.  If  the  ratio  of  two  variables  is  constant,  the 
variables  are  said  to  be  in  direct  variation. 

Thus,  when  m  is  a  constant,  if  -  =  m,  x  varies  directly  as  y] 
ov  xocy,  ^ 


338  ELEMENTARY    ALGEBRA  [Cii.  XXI,  §  259 

The  height  of  a  eohiran  of  mercury  in  a  thermometer  is 
•  known  to  vary  as  the  temperature.     If  //  and  //'  represent 
the  different  heights  of  the  mercury  when  the  temperatures 
are  respectively  Z'and  T\  IIccT-,  or,  II:  II'  =  T :  T', 

2.  If  the  ratio  of  a  variable  to  the  reciprocal  of  a  second 
variable  is  constant,  the  variables  are  said  to  be  in  inverse 
variation. 

Thus,  when  m  is  a  constant,  if  xi—  =  m,  x  varies  inversely 

1  y 

as  y,  or  xcc  — 

y 

The  volume  of  a  gas  is  known  to  vary  inversely  as  the 
pressure.     If  V  and  V  represent  the  volumes  of  a  gas  under 

the  respective  pressures  P  and  P',  Foe—;  or,  F:  F'=—  :  — 
which  may  be  more  conveniently  w^ritten  V:  V  =  P' :  P. 

3.  If  the  ratio  of  a  variable  to  the  product  of  two  other 
variables  is  a  constant,  the  first  variable  is  said  to  be  in 
joint  variation  with  the  other  two  variables. 

Thus,  when  m  is  a  constant,  if  x :  yz  =  m,  x  varies  jointly  as 
y  and  z ;  or,  x  oc  yz. 

The  distance  travelled  depends  npon  the  rate  and  the  time. 
If  D  and  D'  represent  the  distances  travelled  when  the  rates 
and  times  are  respectively  R  and  jR',  T  and  T^DzcPT^  or, 

4.  If  the  ratio  of  a  variable  to  a  second  variable  multi- 
plied by  the  reciprocal  of  a  third  variable  is  a  constant,  the 
first  variable  is  said  to  be  in  direct  and  inverse  variation 
with  the  second  and  third  variables. 

Thus,  when  m  is  a  constant,  if  x  \  (y  -  -A^m^  x  \.^  \\).  direct 

y 
and  inverse  variation  with  y  and  z\  or^  a;  oc  -  • 


Ch.  XXI,  §  260]  VARIATION  339 

The  base  of  a  rectangle  is  known  to  vary  as  the  area  divided 
by  the  altitude.  If  B  and  B^  represent  the  bases  when  the 
areas   and   altitudes   are  respectively  S  and   S\  A  and  A\ 

B«^.l;   or,  £:^'  =  |:|. 


PRINCIPLES   OF  VARIATION 

260.      I.    If  X  <x:  y^  and  y  ^z^  then  xocz. 
When  m  and  n  are  constants,  let 

~  =  m,  or  x  =  ym^  (1) 

y 

and  let  ^-^n,  or  y  =  zn^  (2) 

z 

multiplying  (1)  and  (2),        xy  =  yzmn^  (3) 

X 

dividing  (3)  by  yz^  -  =  mn.  (4) 

z 

In  (4),  since  m  and  n  are  constants,  mn  is  also  a  con- 
stant: hence  xccz. 

II.    If  xccy^  and  x^  oc  y'^  then  xx^  oc  yy'. 
When  m  and  n  are  constants,  let 

^  =  m,  (1) 

and  let  -;  =  w,  (2) 

multiplying  (1)  and  (2),       — ^  =  m/j.  (3) 


Hence  a;^;'  oc  y^'. 
Similarly,  if  xocy^  xT-  cc  y\ 


340  ELEMENTARY   ALGEBRA         [Ch.  XXI,  §  260 

III.  If  xccy^  then  Jcx  x  %. 

Let  k  be  either  a  constant  or  a  variable ;  and  let  m  be  a 

constant.     Let  ^ 

-  =  m,  (1) 

1/ 

multiplying  -  in  (1)  by  -,    t-  =  ^'  (2) 

2/  K     ky 

Hence  kx  oc  ky. 

IV.  If  xQcyz^  then  y  cc  -,  and  zee-. 

z  y 

When  m  is  a  constant,  let 

(1) 

(2) 
(3) 


m 

__   X 

'^  yz 

or  myz^x^ 

dividing  (1)  by  mz^ 

mz 

dividing  (2)  by  -, 
z 

1=1. 

X     m 
z 

Hence              y  oc 

X 

z' 

Similarly,  ^  x  -  • 

y 

V.  If  xoc  y  when  z  is  constant^  and  if  x  ccz  when  y  is 
constant^  then  x  oc  yz  when  both  x  and  y  are  variables. 

Let  X,  y,  z;  x\  y\  z\  a;",  y\  z\  be  three  sets  of  corre- 
sponding values  of  x^  y,  and  z. 

If  z  is  constant,  —  =  -^,  (1) 

^     y 

if  y  is  constant,  --  =  --,  (2) 

multiplying  (1)  and  (2),     4=  4%  .       (3) 

X       y  z 


Cn.  XXI,  §  260] 

VARIATION 

or,  rewriting  (3), 

X          7^' 

yz     y'z^ 

Hence 

XQc  yz. 

341 
(4) 


EXAMPLES 

1.  If  a;  oc  y,  and  if  a;  =  3  when  y  =  2,  find  x  when  y=Q. 

Let                                    -  =  m,  ri) 

y  ^ 

substituting  in  (1)  a:  =  3,  ^  =  2,             I  =  ^9  (2) 

X  3 

substituting  in  (1)  ^  =  6,                 -  =  m  =  -,  (3) 

u  A 

solving  (3),                                                  a;  =  9.  (4) 

2.  If  y  varies  inversely  as  the  square  of  x^  and  if  «/  =  8 
when  a;  =  3,  find  x  when  2/  ==  2. 

Let                          y  =  ^^                    yQ?^m,  (1) 

substituting  in  (1)  ?/  =  8  and  a;  =  3,           72  =  m,  (2) 

substituting  in  (1)  y  =  2  and  m  =  72,     2  oj^  =  72,  (3) 

solving  (3),                                                       x=  ±&.  (4) 

3.  If  a; Qc - ,  and  if  a;  =  4  when  y=Q  and  ^  =  3,  what  is 
the  value  of  x  when  y  =  Q  and  2  =  9  ? 

Let                     -  =  m,                   —  =  m,  (1) 

substituting  in  (1),  a?  =  4,  ?/  =  6,  2:  =  3,        m  =  2,  (2) 

substituting  in  (1),  2/  =  6,  ;s  =  9,  m  =  2,     ^  =  2,  (3) 

solving  (3),                                                     a?  =  |.  (4) 


342  ELEMENTARY   ALGEBRA  [Cn.  XXI,  §  l>r;0 

4.  The  volume  of  a  sphere  varies  as  the  cube  of  the 
radius,  and  the  volume  of  a  sphere  is  1^:37^  when  the 
radius  is  7.  Find  the  volume  of  a  sphere  whose  radius  is 
14. 

Let  "F  represent  the  volume  and  R  the  radius  of  the  sphere. 

Then  '^  =  ^^^'  ^^   V=mE%  (1) 

substituting  in  (1)  F=  14371  and  B  =  7,     m  =  |^  ^  (2) 

hence  volume  =  ^^^  •  14^  =  ^^  •  8  =  11498|.  (3) 

EXERCISE   CXXXIV 

1.  If  a:  Qc  ?/,  and  if  x  =  5  when  ^  =  4,  find  x  when  y  =  9. 

2.  If  xcc  —.  and  if  2:  =  4  when  v  =  3,  find  y  when  x—2. 

3.  li  xoc  yz^  and  if  a;  =  2  when  ^  =  3  and  2  =  4,  find  x 
when  2/  =  2  and  ^  =  6. 

4.  If  X  Qc  -,  and  if  :r  =  16  when  y  =  3  and  2=8,  find  2 
when  a:  =  12  and  t/  =  2. 

5.  If  2:  X  -  +  -,  and  if  a;  =  4  when  y  =  3  and  2  =  5,  find 

^      z 

y  when  :r  =  3  and  2  =  2. 

6.  If  :?;  varies  -  directly  as  y  and  inversely  as  2,  and  is 
equal  to  4  when  2/  =  2  and  2=3,  what  is  the  value  of  x 
when  2/  =  35  and  2  =  15? 

1 ,  \i  y  =.  u  —  v^  a  u  varies  as  x^  and  v  as  a;^,  and  if 
1/  =  2  when  :r  =  1,  and  2/  =  3  when  x  =  2,  find  the  value 
of  y  in  terms  of  x, 

8.  If  a^  —  52  varies  as  c^,  and  if  c=  2  when  a=  5  and 
6  =  3,  find  the  equation  between  a,  J,  and  (?. 


Cii.  xxr,  §  200]  VARIA  rioN  34S 

9.    If  x^y^  and  z  xy,  prove  that  x  —  zocy. 

10.  If  2;  X  2/,  prove  that  x^  +  y'^oc  xy, 

11.  li  x  +  y  OCX  —  y,  prove  that .x!^  +y^Qc xy, 

12.  If  xyzy^  and  xocz,  and  xocw^  when  ^  and  «^^,  y  and 
z^,  2/  iiiid  z,  are  constants,  prove  that  xccyzw. 

13.  The  area  of  a  circle  varies  as  the  square  of  the 
radius ;  show  that  the  area  of  a  circle  of  5  feet  radius  is 
equal  to  the  sum  of  a  circle  of  3  feet  radius  and  another  of 
4  feet  radius. 

14.  Knowing  that  the  volume,  F,  of  a  gas  varies  directly 
as  the  temperature,  T,  when  2^=  273°  + the  number  of 
degrees  in  temperature  (in  the  Centigrade  System):  if 
the  volume  of  a  certain  gas  is  400  c.c.  when  the  tempera- 
ture is  27°  C,  find  the  volume  of  the  gas  at  127°  C. 

15.  Find,  under  the  law  given  in  the  preceding  example, 
the  volume  of  a  gas  at  0°  C,  if  the  volume  is  250  c.c.  at 

18°  C. 

16.  Knowing  that  the  volume,  F  of  a  gas  varies  inversely 
as  the  pressure,  P,  upon  it :  if  the  volume  of  a  gas  is 
100  c.c.  when  the  pressure  is  76  cm.,  find  the  volume  when 
tlie  pressure  is  38  cm. 

17.  Under  the  conditions  given  in  the  preceding  prob- 
lem, if  the  volume  of  a  gas  is  600  c.c.  when  the  pressure 
is  60  cm.,  find  the  pressure  if  the  volume  is  150  c.c. 

18.  Knowing  that  the  intensity  of  illumination,  J,  varies 
inversely  as  the  square  of  tlie  distance,  D :  if  a  candle 
throws  a  certain  amount  of  light  on  a  screen  2  feet  dis- 
tant, what  will  be  its  relative  illuminating  power  at  a 
distance  of  7  feet? 


344  ELEMENTAKY   ALGEBRA  [Ch.  XXI,  §  260 

19.  Under  the  conditions  given  in  the  preceding  prob- 
lem, if  a  candle  and  a  gas  flame  are  12  feet  apart,  and  if  the 
gas  flame  is  equivalent  to  4  candles,  where  must  a  screen 
be  placed  on  a  line  joining  the  candle  and  gas  flame  so  that 
tlie  screen  may  be  equally  illumined  by  each  of  them? 

V  P       V  P 

20.  Knowing  that   -4r^=  -\^  where  F^  V^,  etc.,  are 

as  given  in  Problems  14  and  16 :  if  a  mass  of  air  at  0°  C. 
has  a  volume  of  600  c.c.  at  a  pressure  of  76  cm.,  find  the 
volume  when  the  temperature  is  91°  C.  and  the  pressure 
is  190  cm. 

21.  Under  the  conditions  given  in  the  preceding  prob- 
lem, if  the  volume  of  a  certain  mass  of  air  at  27°  C,  and 
under  a  pressure  of  225  cm.  is  2000  c.c,  find  its  volume 
at  127°  C,  under  a  pressure  of  75  cm. 

22.  Knowing  that  the  amount  of  bending,  5,  of  a  rod 
varies  jointly  as  the  load,  i,  and  the  cube  of  the  length, 
i',  and  inversely  and  jointly  as  the  width,  Tf,  and  the  cube 

of  the  thickness,  T,  that  is,  B  oc  :  if  a  rod  8  feet  long, 

4  inches  wide,  1  inch  thick,  is  bent  0.2  inch  by  a  weight 
of  50  pounds,  how  much  would  a  weight  of  5  pounds  bend 
a  rod  of  like  material,  24  feet  long,  8  inches  wide,  and 
2  inches  thick  ? 

23.  Under  the  conditions  given  in  the  preceding  prob- 
lem, if  a  beam  16  feet  long,  8  inches  wide,  4  inches  thick, 
is  bent  ^  inch  by  a  weight  of  1000  pounds,  how  much 
would  a  beam  10  feet  long,  6  inches  wide,  8  inches  thick, 
be  bent  by  the  same  weight  ? 


CHAPTER  XXII 

PROGRESSIONS 

ARITHMETICAL  PROGRESSION 

261.  A  succession  of  terms,  each  of  which  is  obtained 
from  the  preceding  term  by  the  addition  of  the  same  posi- 
tive or  negative  quantity  (the  common  difference),  is  called 
an  arithmetical  progression. 

Thus,  2,  5,  8,  11,  etc.,  and  —  1,  —  2,  —  3,  etc.,  are  arithmeti- 
cal progressions. 

The  first  term  is  usually  represented  by  a,  and  the  com- 
mon difference  by  d ;  hence  the  progression  is  a,  a  +  6?, 
a  +  2  c?,  a  +  3  c?,  etc.  The  number  of  terms  in  a  progres- 
sion is  represented  by  n  ;  and  the  nth.  term  by  /. 

Since  each  term  is  formed  from  the  preceding  term  by 
the  addition  of  6?,  the  coefficient  of  c?,  in  any  term,  is  one 
less  than  the  number  of  the  term  in  the  progression. 
Thus,  the  third  term  is  a  +  2  d  \  hence 

/  =  a +(/(,! -1).  I. 

1.    Find  the  10th  term  of  the  progression  2,  5,  8,  etc. 

By  the  conditions,      a  =  2,  d  =  3,  ^  =  10, 

by  I,  10th  term  =  2  +  3  (10  - 1)  =  29. 

345 


346  ELEMENTARY   ALGEBRA        [Ch.  XXII,  §  26] 

2.    Find  the  10th  term  of  the  progression  in  which  the 
3d  term  is  11,  and  the  7th  term  is  27. 


By  the  conditions, 

a  +  2d  =  ll, 

(1) 

and, 

a  +  6d  =  27. 

(2) 

subtracting  (1)  from  (2), 

4d!  =  16, 

(3) 

or. 

d=  4, 

(4) 

substituting  c^  =  4,  in  (1), 

a=  3, 

(5) 

by  I,                        10th  term  = 

=  a  +  9  d  =  39. 

(6) 

EXERCISE    CXXXV 

Find  the  last  term  of  each  of  the  following  progressions: 

I.  2,  5,  8,  •••  to  10  terms.        2.   8,  5,  2,    ••  to  It  terms. 

3.  100,  95,  90,  ...  to  15  terms. 

4.  5,  6  —  (?,  J  —  2  (?, ...  to  13  terms. 

Find  the  nth  term  of   the  following   progressions   in 
^^^^^=  5.   a  =  3i,cZ=2f,^=10. 

6.  a  =  76f,  (^=-4f,  n  =  8. 

7.  a  =  h  +  c^  d  =  h  —  0%  n  =  p. 

8.  a=x  —  7/^d=  —  t/^n  =  x^  —  y^. 

Find  the  indicated  terms  in  the  following  progressions : 
9.    7th  term ;  the  3d  being  10,  and  the  10th,  —  5. 
10.   6th  term ;  the  4th  being  0,  and  the  9th,  15. 

II.  1st  term ;  the  7th  being  —  48,  and  the  13th,  —  108. 

12.  10th  term ;  the  5th  being  28,  and  the  9th,  52. 

13.  15th  term  ;  the  31st  being  —  40,  and  the  sum  of  th 
3d  and  11th,  4. 


tne 

i 


Ch.  XXII,  §  262]  PROGRESSIONS  347 

262.  When  three  quantities  are  in  arithmetical  pro- 
gression, the  middle  term  is  called  the  arithmetical  mean 
between  the  other  two. 

If  a,  J,  and  c  are  in  arithmetical  progression,  the  arith- 
metical mean  h  can  be  found  in  terms  of  the  other  two. 
Since  b  —  a  =  c  —  h^  i  =  ^  (a  +  c). 

Hence,  the  arithmetical  mean  between  two  quantities  is 
one-half  the  sum  of  the  quantities. 

In  an  arithmetical  progression  containing  any  number 
of  terms,  all  the  terms  between  the  first  and  last  are  called 
arithmetical  means  between  those  terms. 

Insert  6  arithmetical  means  between  8  and  29. 

The  progression  evidently  contains  8  terms ;  a  =  Sj  n  =  8, 
Z  =  29. 

By  I,  29  =  8 +  (^(8-^1),  (1) 

solving  (1),  cZ=3.  (2) 

Hence  the  progression  is,  8,  [11,  14,  17,  20,  23,  26,']  29. 

EXERCISE   CXXXVI 

1.  Insert  7  arithmetical  means  between  69  and  95. 

2.  Insert  13  arithmetical  means  between  13  and  209. 

3.  Insert  98  arithmetical  means  between  6  and  —  489. 

4.  Insert  99  arithmetical  means  between  —  5780  and  0. 

5.  Insert  4  arithmetical  means  between  k  and — 

6.  Insert  10  arithmetical  means  between  w  +  V3  and 
m  +  V3  +  729. 

7.  Insert  r  arithmetical  means  between  1  and  3  r  —  2. 


348  ELEMENTARY  ALGEBRA        [Ch.  XXII,  §  263 

263.    If  aS^  denotes  tlie  sum  of  n  terms  of  an  arithmetical 
progression, 

^=a  +  (a  +  ^)  +  (a  +  2(?)+...  +  (Z  -d')  +  l,       (1) 

cr,        S^l  +(l  -d)  +  (l  ^2d)+'"+Qa  +  d')  +  a,      (2) 

adding  (1)  and  (2), 

2Ay=(a+Z)  +  (a  +  Z)  +  (a  +  0+-+(^  +  0  +  (^  +  0.      (3) 
or,  2S==n(a  +  l),  (4) 

whence,  5==-[a  +  /)-  II. 

Since,  by  I,  Z=  a  +  d(n  —  1),  substituting  I  in  II, 


5  =  ? 


2  a  +  (/(/?- 1) 


III. 


Equations  I,  II,  and  III   are   called   the   formulas   of 
arithmetical  progression. 

1.  Find  the  sum  of  6  terms  of  the  progression,  5,  3,  1, 
—  1,  etc. 

By  the  conditions,  a  =  5,  (^  =  —  2,  n  =  6. 

Substituting  a,  d,  and  n  in  III,  aS  =  f  f  10  -  2  (5)  |  =  0. 

2.  How  many  terms  of  the   progression,  4,  7,  10,  ••• 
must  be  taken  in  order  that  the  sum  may  be  69? 

By  the  conditions,  a  =  4,  d  =  3,  S  =  69o 

Substituting  a,  d,  and  S  in  III,  69  =  -  1 8  +  3  (ri  - 1)  1 ,     (1) 

reducing  (1),  3n'  +  5n^l3S  =  0,  (2) 

solving  (2),  n  =  6,  or  -  2/.  (3) 

Since  n  must  always  be  a  positive  integer,  n  =  6  is  the  only 
solution. 


Ch.  XXII,  §  263]  PROGRESSIONS  349 

Problems  of  tlie  class  stated  above  will  evidently  always 
involve  the  solution  of  a  quadratic  equation,  and  it  is  therefore 
possible  to  obtain  one,  two,  or  no  correct  solutions  according  as 
one,  two,  or  no  solutions  of  the  quadratic  equation  are  positive 
integers. 

3.  In  an  arithmetical  progression  whose  first  term  is  3, 
the  sum  of  7  terms  is  105.     Find  the  common  difference. 

By  the  conditions,  a  =  3,  n  =  7,  S  =  105. 

Substituting  a,  n,  and  S  in  III,  105  =  |-(6  +  6  d),  (1) 

solving^),  cZ  =  4. 

EXERCISE    CXXXVII 

Find  the  sum  in  each  of  the  following  progressions : 

1.  1,  2,  3,  4,  ...  to  10  terms.        3.    7,  17,  27,  ...  to  8  terms. 

2.  |-,  J,  ^,  ...  to  12  terms.  4.    2,  2|,  3|-,  ...  to  m  terms. 

5.  6|,  9|l  121^,  ...  to  13  terms. 

6.  100,  90,  80,  ...  to  21  terms. 

7.  178,  171,  164,  ...  to  11  terms. 

8.  1,  1  +  V2,  I+2V2,  ...  to  r  terms. 

Find  the  number  of  terms  in  each  of  the  following  pro- 
gressions, so  that  the  given  sum  may  be  obtained : 

9.  aS'=45;  15,  12,  9,  .... 

10.  aS^=-1545;  50,43,  36,  .... 

11.  /S^=1200;   31,  38,45,  .... 

12.  ^=52i;   |,  |,  1,  .... 

13.  jS  =  80(1+W2);  3-V2,  3,  3  +  V2,  .... 


350  ELEMENTARY  ALGEBRA        [Ch.  XXII,  §  263 

In  the  following  arithmetical  progressions : 

14.  Find  c?,  and  Z,  if  a  =  3  and  the  sum  of  the  first  13 
terms  is  351. 

15.  Find  6?,  if  the  12tli  term  is  38  and  the  sum  of  the 
first  13  terms  is  351. 

16.  Find  d,  and  Z,  if  a  =  222,  n  =  223,  and  jS=^0. 

17.  Find  a,  and  l,itd  =  Q,n  =  10,  and  S=  310. 

18.  Find  n,  and  c?,  if  ^  =  4,  Z  =  —  22,  and  S==  —  99. 

19.  Find  ?^,  and  cZ,  if  a  =  ^,  Z  =  15|^,  and  S  =  47. 

20.  Find  71,  and  a,  if  cZ  =  a;  —  1,  I  =  a^  +  x^  +  S  x  —1^  and 
S=5a^  +  Sx^  +  6x. 

21.  The  sum  of  the  first  6  terms  is  261,  and  the  sum 
of  the  first  9  terms  is  297.     Find  the  first  9  terms. 

22.  The  sum  of  the  first  3  terms  is  14,  and  the  sum  of 
the  squares  of  these  terms  is  78.     Find  the  terms. 

23.  The  sum  of  the  first  half  of  the  terms  is  28,  the 
sum  of  the  second  half  is  222,  the  sum  of  the  first  and 
last  terms  is  50.     Find  the  number  of  terms. 

24.  The  sum  of  the  last  four  terms  is  20,  the  product 
of  the  second  and  fifth  is  16.  If  the  progression  contains 
five  terms,  find  the  progression. 

25.  In  a  progression  of  eighteen  terms  the  product  of 
the  two  middle  terms  is  90,  and  the  product  of  the  first  and 
eighteenth  terms  is  18.     Find  the  first  and  last  terms. 


Ch.  XXII,  §  264J  PROGRESSIONS  351 

GEOMETRICAL  PROGRESSION 

264.  A  succession  of  terms,  each  of  which  is  obtained 
from  the  preceding  term  by  multiplying  it  by  the  same 
positive  or  negative  quantity  (the  common  ratio),  is  called 
a  geometrical  progression. 

Thus,  2,  4,  8,  16,  etc.,  and  1,  —  3,  9,  —  27,  etc.,  are  geomet- 
rical progressions. 

The  first  term  is  usually  represented  by  a,  and  the 
common  ratio  by  r;  hence  the  progression  is,  a  +  ar  +  ar^ 
+  ar^,  etc.  The  number  of  terms  in  a  progression  is  rep- 
resented by  /?,  and  the  ^th  term  by  /. 

Since  each  term  is  formed  from  the  preceding  term  by 
multiplying  it  by  r,  the  exponent  of  r  in  any  term  is  one 
less  than  the  number  of  the  term.  Thus,  the  third  term 
is  ar^;  and  the  72-th  term  or 

/  =  ar^-i.  I. 

1.  Find  the  7th  term  of  the  progression  1,  —  3,  9,  •••. 

By  the  conditions,    a  =  1,  r  =  —  3,  n  =  7, 
by  I,'  7th  term  =  1(- 3)^  =  729. 

2.  If  the  4th  term  of  a  geometrical  progression  is  1, 
and  the  7th  term  is  ^-,  find  the  1st  term. 

By  the  conditions,  m^  =  1,  (1) 

and,  ar^  =  i,  (2) 

dividing  (2)  by  (1),  7^  =  ^,  (3) 

from  (3),  r  =  i,  (4) 

substituting  r  =  |  in  (1),  a  =  8.  (5) 


352  ELEMENTARY  ALGEBRA        [Ch.  XXII,  §  264 

EXERCISE    CXXXVIII  J| 

Find  the  last  term  in  each  of  the  following  geometrical 
progressions : 

1.  2,  6,  18,-..  to  7  terms.  4.    27,  9,  3,  ...  to  8  terms. 

2.  3,  —  6, 12,  ...  to  6  terms.     5.    6,  3,  |,  ...  to  10  terms. 

3.  4,  8,  16,  ...  to  7  terms.        6.    1,  -f,  \^-,  ...  to  11  terms. 

In  the  following  geometrical  progressions : 

7.  Find  the  7th  term,  the  2d  term  being  75,  and  the 
5th,  -f. 

8.  Find  the  2d  term,  the  4th  term  being  —  5,  and  the 
7th,  625. 

9.  Find  the  15th  term,  the  5th  term  being  -2^,  and  the 

Q8  • 

10th,  -. 

2^ 

10.  Find  the  50th  term,  the  19th  being  1200,  and  the 
29th,  1200. 

11.  Find  the  11th  term,  the  2d  term  being  P  —  c^,  and 
the  5th,  (b  +  c)(b-cy. 

12.  Find  the  10th  term,  the  3d  term  being  h^^  and  llie 

13.  Find  the  7th  term,  the  2d  term  being  1,  and  th^. 
4th,  17  -  12  V2. 

14.  Find  the  8th  term,  the  4th  term  being  49  —  20  V6, 
and  the  6th,  485  -  198  V6. 

15.  Find  the  7th  term,  the  3d  term  being  —  2,  and  the 
8th,  ~2i. 


Cii.  XXII,  §  265]  PROGRESSIONS  353 

265.  When  three  quantities  are  in  geometrical  progres- 
sion, the  middle  term  is  called  the  geometrical  mean 
between  the  other  two. 

If  a^  5,  and  c  are  in  geometrical  progression,  the  geo- 
metrical mean,  which  is  a  mean  proportional,  can  be  found 

in  terms  of  the  other  two.     Since  -  =  7, 

a      0 

lfl=ac,  (1) 

extracting  square  roots  in  (1),        h  =  -Vae.  (2) 

Hence,  the  geometrical  mean  between  two  quantities  is  the 
square  root  of  the  product  of  those  quantities. 

In  a  geometrical  progression  containing  any  number  of 
terms,  all  the  terms  between  the  first  and  last  are  called 
geometrical  means  between  those  terms. 

Insert  3  geometrical  means  between  6  and  486. 

The  progression  evidently  contains  5  terms ;  a  =  6,  n  =  5, 
Z  =  486. 

By  I,  486  =  6  r\  (1) 

solving  (1),  r  =  3.  (2) 

Hence  the  progression  is  6,  [18,  64,  162,]  486. 

EXERCISE    CXXXIX 

1.  Insert  2  geometric  means  between  1  and  64. 

2.  Insert  6  geometric  means  between  —  and ^• 

3.  Insert  11  geometric  means  between  1  and  2. 

4.  Insert  5  geometric  means  between  1875  and  3. 

5.  Insert  5  geometric  means  between  36  and  —  -^^-^c 


354  ELEMENTARY  ALGEBRA        [Cii.  XXII,  §  2G6 

266.    If  S  denotes  the  sum  of  n  terms  of  a  geometrical 
progression, 

S=  a  +  ar  +  ar^  +  ar^  +  •••  ar'*"^  +  ar^'^.         (1) 

Multiplying  (1)  by  r, 

rS=  ar  +  ar^  +  ar^  -{-  ar^  +  •••  ar^~^  +  ar%  (2) 

subtracting  (1)  from  (2), 

8(r  -  1)  =  ar''  -  a,  (3) 

from  (3),  5  =  ^'^^^^  =  ^SllnSl.  H. 

r  —  1  /•—  1 

Since  Z  =  ar^~^^  rl  =  ar**,  substituting  rl  for  ar**  in  II, 

5=^.  III. 

r—  1 

1.  Find  the  sum  of  the  progression,  2,  6,  18,  •••  to  6 
terms. 

By  the  conditions,  a  =  2,  r  =3,  n  =  6. 

By  II,  /y  =  ^(y7^  =  728. 

2.  The  3d  term  of  a  geometrical  progression  is  27,  the 
6th  is  81.     Find  the  sum  of  the-  first  5  terms. 

By  the  conditions,  ar  =  27,  (1) 

and,  ar^  =  81,  (2) 

dividing  (2)  by  (1),  t'=   3,  (3) 

from  (3),  r  =  V3,  (4) 

substituting  in  (1),  a  =  9,  (5) 

substituting  in  II,  S  =  ^(^^f-^  =  117  +  36-/3,     (6) 

V3-1 


Cii.  XXII,  §  266]  PROGRESSIONS  '  355 

EXERCISE    CXL 

In  the  following  geometrical  progressions : 

1.  Find  the  sum  of  3,  —  6,  12,  •••  to  6  terms. 

2.  Find  the  sum  of  6,  |,  |,  •••  to  10  terms. 

3.  Find  the  sum  of  |,  |,  |,  •••  to  10  terms. 

4.  Find  the  sum  of  V2,  2,  2V2,  ...  to  8  terms. 

5.  Find  the  sum  of  V2  +  1,  1,  V2  —  1,  ...  to  5  terms. 

6.  Find  the  sum  of  the  first  7  terms,  if  the  2d  term 
Ls  4,  and  the  5th,  256. 

7.  Find  the  sum  of  the  first  5  terms,  if  the  3d  term 

is  27,  and  the  5th,  48. 

8.  If  a  =  6,  and  r  =  —  2,  find  n^  if  the  sum  of  n  termii 
is  -30 

9.  The  sum  of  the  first  5  terms  is  242,  and  the  com- 
mon ratio  is  3.     Find  the  5th  term. 

10.  The  sum  of  the  first  4  terms  is  9|^,  and  the  common 
ratio  is  \,     Find  the  1st  term. 

11.  Find  the  sum  of  the  first  6  terms,  if  the  6th  term 
is  —  ^Ys  ^^^  ^^^  common  ratio  is  —  |. 

12.  Find  the  common  ratio,  and  the  sum  of  the  first 
5  terms,  if  the  1st  term  is  \  and  the  6th  term  is  864. 

13.  Find  the  sum  of  the  first  10  terms  of  a  geometric 
progression  in  which  the  1st  term  is  243  and  the  common 

ratio  is • 

V3 

14.  If  the  4th  term  is  y^g,  and  the  7th  term  is  yj^,  how 
many  terms,  beginning  with  the  1st,  must  be  taken  so 
that  their  sum  is  ||||  ? 


356  ELEMENTARY  ALGEBRA     [Ch.  XXII,  §§  267,268 

267.  As  in  §  258,  if  a  quantity  retains  the  same  value 
throughout  a  particular  investigation,  it  is  called  a  con- 
stant. If  a  quantity  changes  in  value  during  a  particu- 
lar investigation,  it  is  called  a  variable. 

When  the  value  of  a  variable  can  be  made  to  approach 
the  value  of  a  constant  in  such  a  way  that  tPie  difference 
of  the  variable  and  the  constant  can  be  made  less  than 
any  assigned  quantity,  however  small,  the  constant  is 
called  the  limit  of  the  variable. 


SUM   OF   AN  INFINITE   GEOMETRICAL  PROGRESSION 

268.  If  r>l,  each  term  of  a  geometrical  progression  is 
larger  than  the  preceding  term,  and  the  sum  of  n  terms 
must  increase  indefinitely  as  n  increases.  If  r  =  1,  the 
terms  are  all  equal,  and  the  sum  of  n  terms  must  again 
increase  indefinitely  as  n  increases.  If  r  <  1,  and  r  >  —  1, 
each  term  is  less  than  the  preceding  term  ;  and  it  will  be 
seen  that  the  sum  of  n  terms  always  remains  less  than 
some  definite,  finite  quantity  ;  from  which,  however,  by 
increasing  n^  it  can  be  made  to  differ  by  less  than  any 
assigned  quantity,  however  small. 

As  an  illustration,  consider  the  geometrical  progression, 
1  +  1  +  1  ....      Applying    III,    S=^~'^^  =  2-l,    Hence,   in 

this  progression,  the  sum  of  any  number  of  terras  differs 
from  2  by  just  the  last  term.  But,  by  increasing  n  the  last 
term  can  be  made  as  small  as  may  be  required.  Evidently 
the  sum  of  n  terms  can  never  be  as  large  as  2,  but  it  can 
be  made  to  differ  from  2  by  a  quantity  less  than  any  as- 
signed value.  Hence  2  is  the  limit  of  the  sum  of  n  terms, 
as  n  increases  indefinitely. 


Ch.  XXII,  §  269]  PROGRESSION  357 

269.    When  r  <1,  it  is  convenient  to  write  II  in  the 
form, 


1  —  r        1  —  rl  —  r 

Here  f^  can  be  made  as  small  as  is  required  by  increas- 
ing n.     The  second  fraction, ,  can,  therefore,  be  made 

1  —  r 

as  small  as  is  required  by  increasing  the  number  of  terms; 
and  S  can  be  made  to  differ  from by  less  than  any  as- 
signed quantity. is,  therefore,  the  limit  approached 

by  aS^  as  ^  increases  indefinitely.     It  is  usually  called  the 
%nm  of  the  infinite  geometrical  progression,  but  this  must 
always  be  understood  to  mean  the  limit  of  the  sum  of  the 
progression  as  n  increases  indefinitely. 
If  S  represents  the  limit  of  that  sum, 

IV. 


1-r 


1.  Find  the  sum  of  an  infinite  number  of  terms  in  the 
progression,  l,  1  -^^,  etc. 

By  IV,  ^=_J_=?. 

2.  Find  the  value  of  0.4545  .... 

The  decimal  0.4545  is  evidently  the  geometric  progression 

45      I  4  5       _i  4  5  _i_ 

TTJU  "T  T  0  0  0  Ty  ~t"  TTTTJ  0  dTFTT  "T   *  *  *> 

I   in  which  «  =  tVV  ^'  =  Tk- 

t 

By  IV,  ^  =  ^=5. 


358  ELEMENTARY   ALGEBRA        [Ch.  XXII,  §  269 

3.   Find  the  value  of  0.4555  .... 

The  decimal  0.4555  is  evidently  y\  +  the  progressicn 

ron  +  rtTo  +  Tiro  ¥"0"  +  •••; 
in  which  a  =  -^-^,  r  =  -^^^ 

By  IV,  ^=^  =  1.  (1) 

1  -  tV     J-^ 

Hence        0.4555  ...  =3-\  +  J^  =  |i.  (2) 

EXERCISE    CXLI 

In  the  following  infinite  geometrical  progressions : 

1.  Sum  to  infinity,  2,  —  f,  |,  .••. 

2.  Sum  to  infinity,  5,  21,  1^,  ••.. 

3.  Sum  to  infinity,  3|,  —  2|^,  11,  •••. 

4.  Sum  to  infinity,  4,  —  |,  |,  •••• 
6.   Find  the  value  of  0.2544  .... 

6.  Find  the  value  of  0.86464  .... 

7.  Find  the  value  of  0.5124545  .... 

8.  Find  the  value  of  0.2162525  .... 

9.  Find  the  value  of  0.1248248  .... 
10.  Find  the  value  of  0.18301830  .... 

11.  Find  the  sum  to  infinity,  if  the  4th  term  is  36  and 
the  7th  is  - 10|. 

12.  Find  the  1st  term,  if  the  sum  to  infinity  is   —  If 
and  the  2d  term  is  2. 

13.  Find  the  4th  term,  if  the  1st  term  is  100  and  the 
sum  to  infinity  is  lll^. 


Cii.  XXIT,  §  270]  PROGRESSION  *  359 

270.  A  succession  of  quantities,  whose  successive  terms 
are  arranged  in  accordance  witli  some  law,  is  called  a  series. 

Thus,  arithmetical  and  geometrical  progressions  are 
series. 

If  a  series  of  quantity  be  given,  it  must  be  tested  to 
determine  the  nature  of  the  series. 

The  abbreviations  A.  P.  and  G.  P.  indicate  respectively 
arithmetical  and  geometrical  progression. 

REVIEW    EXERCISE    CXLII 

1.  Show  that  2a2(a  +  3J),  (a +5)3,  and  2J2(j  +  3^), 
are  in  A.  P. 

2.  How  many  terms  of  the  series  1,  8,  5,  7,  •••  amount 
to  1,234,321  ? 

3.  The  arithmetic  mean  between  two  quantities  is  •^^, 
and  the  geometric  mean  is  2.     Find  the  quantities. 

4.  Find  the  sum  of  the  terms  in  the  series  1,  1  +  J, 
1  +  2  6,  1  +  3  J,  •••  1  +  ^J,  when  5  =  2,  n^ll. 

5.  Sum  the  series  —  3,  6,  first  as  G.  P.,  then  as  A.  P., 
each  to  5  terms. 

6.  If  the  arithmetic  mean  between  a  and  b  be  double 
the  geometric  mean,  find  -• 

0 

7.  How  many  terms  of  the  series  42,  39,  36,  •••  make 
315  ? 

8.  Find  the  sum  of  16  terms  of  the  series  27  +  22| 
+  18+131  +  .... 

7h  — •  1   9?i  ■""  2 

9.  Find  the  sum  of  k  terms  of  the  series  1, , -, 

g  n        n 

n—  3 

,  •••„ 

n 


360  ELEMENTARY   ALGEBRA        [Ch.  XXII,  §  270 

10.  If  a,  5,  c?,  and  d  are  four  quantities  in  G.P.,  show 
that  6  +  c  is  the  geometric  mean  between  a  +  6  and  c  -j-  d. 

11.  Find  the  sum  of  all  integral  numbers  between  1 
and  207,  which  are  divisible  by  5. 

12.  Find  the  sum  of  all  odd  integral  numbers  between 
74  and  692. 

13.  How  many  positive  integral  numbers  of  three  digits 
are  there  which  are  divisible  by  9  ?     Find  their  sum. 

14.  Find  four  numbers  in  A.  P.,  such  that  the  sum  of 
their  squares  shall  be  120,  and  that  the  product  of  the  first 
and  last  terms  shall  be  less  than  the  product  of  the  other 
two  by  8. 

15.  Find  a  G.  P.  in  which  the  sum  of  the  first  two 
terms  is  2,  and  the  sum  to  infinity  is  4. 

16.  The  1st  term  of  an  A.  P.  is  2,  and  d  =  ^.  How 
many  terms  must  be  taken  that  their  sum  amounts  to  192  ? 

17.  Find  the  G.P.  whose  sum  to  infinity  is  4,  and 
whose  second  term  is  J. 

18.  The  sum  of  three  numbers  in  A.  P.  is  —  3,  and 
their  product  is  8.     Find  the  numbers. 

19.  Prove  that  in  an  A.  P.  of  a  limited  number  of  terms, 
the  sum  of  two  terms,  equally  distant  from  the  end  terms, 
is  equal  to  a  constant. 

20.  Prove  that  if  each  term  of  an  A.  P.  be  multiplied 
by  the  same  quantity,  the  resulting  series  will  be  in  A.  P. 

21.  Prove  that  in  a  G.  P.  of  a  limited  number  of  terms, 
the  product  of  two  terms,  equally  distant  from  the  end 
terms,  is  constant. 


N 


Cii.  XXll,  §  270]  PROGRESSION  361 

22.  A  body  slides  down  an  inclined  plane  1290  feet  long 
in  15  seconds.  If  it  slides  9  feet  the  first  second,  and 
thereafter  gains  in  distance  traversed  a  fixed  amount  each 
second,  find  this  gain. 

23.  A  man  deposits  money  in  a  bank  every  week-day 
for  two  weeks.  Tlie  first  day  he  deposits  $1.50,  and  on 
each  succeeding  day  deposits  three  times  as  much  as  on 
the  day  previous.  Find  the  amount  to  his  credit  at  the 
end  of  the  two  weeks. 

24.  In  starting  an  engine  it  was  observed  that  the  fly- 
wheel made  f  of  a  revolution  the  first  second,  3|  revolu- 
tions the  second  second,  and  18|  revolutions  the  third 
second.  If  it  continued  to  gain  speed  at  this  rate,  how 
many  revolutions  would  it  make  in  the  eighth  second  ?  If 
the  wheel  has  a  diameter  of  seven  feet,  how  far  would  a 
point  in  its  rim  travel  in  nine  seconds  ? 

25.  During  a  freshet  the  overflow  pipe  of  a  reservoir 
discharged  in  a  certain  number  of  hours  1,562,496  gallons. 
If  it  discharged  during  the  first  hour  16  gallons  and  it 
continued  to  discharge  on  each  succeeding  hour  five  times 
as  much  as  on  the  hour  previous,  find  the  number  of  hours 
the  overflow  continued  to  increase  and  the  amount  dis- 
charged the  last  hour. 


CHAPTER   XXIII 

PERMUTATIONS   AND   COMBINATIONS 

271.  The  various  orders  in  which  a  number  of  things 
can  be  arranged  are  called  their  permutations. 

Thus,  a  and  h  can  be  arranged  db,  ha]  while  a,  h,  and  c,  can 
be  arranged  ahc,  acb,  hac,  bca,  cab,  cba. 

272.  The  various  groups  that  can  be  selected  out  of 
a  number  of  things,  without  reference  to  their  order,  are 
called  their  combinations. 

Thus,  the  groups  of  two  things  that  can  be  selected  from 
a,  by  and  c,  are  ab,  ac,  and  be. 

Unless  the  contrary  is  expressly  stated,  the  things  whose 
permutations  or  combinations  are  required  will  be  understood 
as  different  things. 

Thus,  the  number  of  permutations  of  three  different  things, 
when  taken  two  at  a  time,  may  be  required. 

273.  //  a  single  operation  can  be  done  in  m  different  ways^ 
and  when  this  operation  has  been  done^  if  a  second  operation 
can  he  done  in  n  different  ways^  the  two  operations  can  he 
done  together  in  mn  different  ways. 

With  the  first  way  of  performing  the  first  operation 
there  may  be  associated  any  one  of  the  n  ways  of  per- 
forming the  second  operation  ;  with  the  second  way  of 
performing  the  first  operation  there    may  be   associated 

362 


Cii.  XXILI,  §  274]     PERMUTATIONS  AND  COMBINATIONS        363 

any  one  of  the  n  ways  of  performing  the  second  opera- 
tion, etc.  That  is,  with  each  one  of  the  m  different 
ways  of  performing  the  first  operation  there  may  be 
associated  n  ways  of  performing  the  second  operation. 
Therefore  there  are  mn  different  ways  of  performing 
the  two  operations. 

Thus,  the  offices  of  president  and  vice-president  can  be 
filled  from  five  candidates  in  20  ways ;  since  any  one  of  the 
five  can  be  selected  for  president,  the  office  of  president  can 
be  filled  in  five  different  ways ;  when  the  office  of  president 
has  been  filled,  any  one  of  the  remaining  four  candidates  can 
be  selected  for  vice-president.  Any  one  way  of  the  five  ways 
of  filling  the  office  of  president  can  be  associated  with  any  one 
way  of  the  four  ways  of  filling  the  office  of  vice-president. 
Therefore  the  two  offices  may  be  filled  in  5  •  4  =  20  different 
ways. 

Similarly,  the  above  principle  applies  to  more  than 
two  operations,  each  one  of  which  can  be  performed  in 
a  definite  number  of  ways. 

Thus,  if  a  man  has  5  coats,  3  waistcoats,  and  6  pairs  of 
trousers,  he  can  dress  himself  in  5  •  3  •  6  =  90  different  ways. 


PERMUTATIONS 

274.  The  number  of  permutations  of  n  different  things 
taken  r  at  a  time  is  n(n  —  !)(/?  —  2)  •••  (n  —  r  +  1). 

The  problem  of  computing  the  number  of  permutations 
Df  n  different  things  taken  r  at  a  time  is  equivalent  to 
bhe  problem  of  filling  r  different  places  with  n  different 
things. 


364 


ELEMENTARY   ALGEBRA       [Cii.  XXIII,  §  275 


The  first  place  can  evidently  be  filled  with  any  one  of 
the  n  different  things.  After  the  first  place  has  been 
filled  there  remain  n  —  1  different  things,  any  one  of 
which  can  be  put  into  the  second  place  ;  that  is,  the 
second  place  can  be  filled  in  n  —  1  different  ways  for 
each  way  that  the  first  can  be  filled.  Hence  the  first 
two  places  can  be  filled  in  n(n  —  1)  different  ways. 

After  filling  the  second  place,  there  remain  n  —  2  dif- 
ferent things,  any  one  of  which  can  be  put  into  the 
third  place ;  that  is,  the  third  place  can  be  filled  in 
n—2  different  ways.  Hence  the  first  three  places  can 
be   filled  in  n(n—  V)(n  —  2)  different  ways,  etc. 


Place      

1st 

2d 

3d 

4th 

... 

rth 

Nurrber  of  ways      .     . 

n 

n-1 

n~-2 

n-3 

... 

n-(r-l) 

Continuing  the  process,  it  is  evident  that  the  number 
of  ways  in  which  each  place  can  be  filled  is  found  by  sub- 
tracting from  n  that  number  which  is  one  less  than  the 
number  of  the  place.  Hence  the  rth  place  can  be  filled 
in  n—  (r  —  X)  =  n  —  r+\  different  ways.  Therefore  the 
r  different  places  can  be  filled  by  n  different  things  in 
n(n  —  V)(n  —  2^  •••  (n  —  r+V)  different  ways. 

The  symbol  for  the  number  of  permutations  of  n  dif- 
ferent things  taken  r  at  a  time  is  written  „P^.     Hence 

./',  =  /7(/i-l)(/7-2)...(/7-r  +  2)(/i-r  +  l).         I. 

275.  The  number  of  permutations  of  n  different  things 
taken  n  at  a  time  can  evidently  be  found  by  substituting 
n  for  r  in  I, 


Cj]  XXIII,  §275]     PERMUTATIONS  AND  COMBINATIONS        365 

„/'„  =  /7(/;-l)(n-2)  •••  (2)(1).  11. 

The  product  of  the  factors  of  „P„,  that  is,  the  product 
of  the  first  n  integral  numbers,  is  called  factorial  /?,  and 
is  written  \novn\     Formula  II  may  therefore.be  written 

„P„  =  n!  II. 

EXAMPLES 

1.  In  how  many  ways  can  8  different  letters  be  inserted 
in  3  different  letter-boxes,  one  and  only  one  being  placed 
in  each  box  ? 

The  first  letter-box  can  be  filled  in  8  different  ways;  the 
second  in  7  different  ways ;  the  third  in  6  different  ways ;  and 
the  three  in  8  •  7  •  6  =  336  different  ways.     That  is, 

by  I,  8P3  =  8(8-1)...  (8-3-M), 

=  8.7.6  =  336. 

2.  In  how  many  ways  can  the  letters  of  the  word 
Pingry  be  arranged  ? 

Since  there  are  6  different  letters,  the  6  different  letters  may 
be  arranged  in  the  6  different  places  occupied  by  the  letters  in 
6 !  different  ways  ;  or, 

by  II,  eP6  =  6!=720. 

3.  In  how  many  different  ways  can  5  people  be  seated 
at  a  round  table  ? 

The  order  of  arrangement  cannot  be  that  of  position  on 
a  straight  line,  but  on  a  closed  curve.  If  one  of  the  5  be 
seated,  so  as  to  give  a  starting-point  from  which  to  reckon  the 
order,  the  remaining  4  can  be  seated  in  the  remaining  4  places 
in  4  !  different  ways  j  or, 

by  II,  4P,  =  4!=24. 


366  ELEMENTxVKY   ALGEBRA     [Ch.  XXIII,  §§  276,  277 

276.  Tlie  number  of  combinations  of  n  different  things 

.       .    /7(/7-l)(/7-2).  .(//-r-f  1) 

taken  r  at  a  tune  is — -- 

rl 

The  symbol  for  the  number  of  combinations  of  n  dif- 
ferent things  taken  r  at  a  time  is  written  nC^- 

Each  one  of  the  combinations  of  JJ^  is  a  selection 
of  r  different  things  which  can  be  arranged,  by  II,  in 
rl  different  ways;  hence  the  number  of  combinations  of  n 
different  things  taken  r  at  a  time,  or  ^(7^,  when  multiplied 
by  rl  equals  the  number  of  permutations  of  ^Prl  that  is, 

nC^'  rl  =7i(n  —  1)  •••  (n  —  r  +  1), 

^       n(n-l}(n-2)-'(n-r  +  l) 

or  n^f—  i  •  ^^^' 

p  l 

The  combinations  of  a,  5,  and  <?,  taken  two  at  a  time, 
are  ab^  ac^  and  be.  Each  combination  can  be  arranged  in 
two  different  ways.     Hence  ^P^  =  fi^  •  2  ;  or 

n   _-  3^2  _  ^  •  ^  _  q 

277.  Formula  III  is  employed  in  obtaining  arithmetical 
results,  but  the  better  form  of  III  for  algebraic  use  is 

c  -—HL—,  IV 

"^  —  — — — .  1  V  . 

r !  (/7  —  r) ! 

Since  by  III, 


n^r 


^  n(n  —  1)(^  —  2)  •••  (n  — r  +  1) 


rl 


n(n  —  \^Qn  —  2)  •••  Qn  —  r  +  1)    (ti  —  r)! 
r\  (n  —  r^l 


By 


There^  <//• 


5* 


^low  m./       EXAMPLES 


1.    How  ,^  ^^■"'"ittees  of  4  men  can  be  formed  from . 


10  men  ? 


Four  men  are  to  be  seleo^^J'^^  ^^  men;  hence,  by  III 

10.9.8:'r 


^"^"'1.2.3.4  ~-^  "'^--< 

2.  From  11  men  find  how  many  committees  of  4  men 
can  be  selected,  when  one  man  is  always  included  on  the 
committee. 

Since  one  man  is  always  included  on  the  committee,  the 

problem  is  to  select  3  men  from  the  remaining  10 ;  hence,  by 

III, 

10-9^^^,^^ 

''  '      1.2.3 

3.  From  9  men  find  how  many  committees  of  3  men 
can  be  selected,  when  one  man  is  always  excluded  from 
the  committee. 

Since  one  man  is  always  excluded  from  the  committee,  the 
problem  is  to  select  3  men  from  the  remaining  8  men;  hence, 

^y"^'  8   7   r 

3(73  =  ^^-11^  =  56. 
*  '     1.2.3 


12  ;,,a,evs?      ;*'"  '""«  «,  ,,  „,„„^  ^^  _^_^^  ^         , 

>jine  men  are'  and  1 

?*^^  '^^^^ted  from  12  n,en-l 

IV,  ^">en;  ],ence,bjin        . 

12  •  1-1     ■  .  ^/> 

OA  a.     Se  J 

6  •  5 

TTT  i-'^'^  ^'  -^^^  ^^  chosen  in  gCs  =  =  15  ways. 

ajy  §  273,  the  entire  committee  can  be  chosen  in  56-15 
=  840  different  ways. 

6.  If  letters  in  any  order  form  a  word,  how  many 
words  can  be  formed  from  8  consonants  and  5  vowels, 
each  word  consisting  of  4  consonants  and  3  vowels  ? 

By  III,  the  selections  of  consonants  and  vowels  are  respec- 
tively gCi  and  sCg. 

'  '     1.2.3.4         ' 

5_,£^^,,^ 
'  '     1.2.3 

By  §  273,  the  total  number  of  selections  of  consonants  and 
vowels  is  70  .  10  =  700.  Since  each  of  the  700  combinations 
consists  of  7  different  letters,  each  combination  can  be  per- 
muted in  7!  =  5040  different  ways.  There  are  700.5040 
=  3,528,000  different  words. 


Cii   X:XI1I,§277]     PERMUTATIONS  AND  COMBINATIONS        369 


EXERCISE    CXLIII 

bind  the 

values  of : 

^'     10^3' 

5.    ,P,. 

2-   n^r 

6.     „Pa- 

^'     12^6- 

7.    ,0,. 

9.    sO,. 

10.     12(78. 

11-     12^10- 
4.     ^Pg.  8.     ^CY  12.     i5(7i4. 

13.  In  how  many  ways  can  10  people  sit  in  4  chairs  ? 

14.  In  how  many  ways  can  the  first  4  letters  of  the 
alphabet  be  arranged  ? 

15.  How  many  numbers  of  3  digits  each,  no  digit  being 
repeated,  can  be  formed  from  the  digits  1  to  9  inclusive  ? 

16.  In  how  many  different  ways  can  2310  be  written  as 
the  product  of  its  prime  factors  ? 

17.  A  man  has  n  different  books,  which  he  can  place  in 
5040  different  arrangements.     Find  the  number  of  books. 

18.  How  many  combinations  can  be  made  of  10  differ- 
ent things  taken  in  sets  of  7  ? 

19.  On  how  many  nights  can  a  different  guard  of  5 
men  be  selected  from  a  body  of  20  ?  On  how  many  of 
these  ^"^iuld  any  one  man  serve  ? 

20.  There  are  20  things  of  one  kind,  and  10  of  another. 
How  many  different  sets  can  be  made  ea,ch  containing  3 
of  the  first  kind  and  2  of  the  second  ? 

21.  In  an  examination  paper  of  10  questions  any  3  can 
be  omitted.     Find  the  number  of  selections. 

22.  In  how  many  ways  can  5  people  form  a  ring  ?  In 
how  many  ways  a  line  ? 


370  ELEMENTARY   ALGEBRA       [Ch.  XXIII,  §  277 

23.  How  many  different  committees  of  3  Republicans 
and  3  Democrats  can  be  formed  from  10  Republicans  and 
7  Democrats  ? 

24.  How  many  even  numbers  of  4  digits  each,  no  digit 
being  repeated,  can  be  formed  from  the  digits  1  to  9 
inclusive  ? 

25.  In  a  boat's  crew  of  8  men  one  man  can  row  only 
on  stroke  side.     How  many  ways  can  the  crew  be  seated  ? 

26.  In  how  many  different  ways  can  a  ball  nine  be 
arranged,  the  pitcher  and  catcher  being  always  the  same, 
but  the  others  playing  in  any  position  ? 

27.  How  many  different  sums  of  money  can  be  formed 
with  a  cent  piece,  a  nickel,  a  dime,  a  quarter,  and  a  half- 
dollar  ? 

28.  How  many  different  quantities  of  anything  ponder- 
able can  be  weighed  with  n  different  weights  ? 

29.  How  many  changes  can  be  rung  with  3  bells  out  of 
6  different  bells  ?     How  many  with  the  whole  peal  ? 

30.  From  100  men  how  many  juries  of  12  men  each  can 
be  selected  if  25  men  are  excused  and  if  A  is  always 
included  ?  ^ 

31.  If  letters  in  any  order  form  a  word,  how  many 
words  can  be  formed  from  7  consonants  and  5  vowels, 
each  word  containing  3  consonants  and  3  vowels,  and 
ending  in  a  consonant  ? 

32.  If  each  of  n  straight  lines  intersects  all  the  others, 
not  more  than  2  lines  intersecting  in  the  same  point,  how 
many  points  of  intersection  will  there  be  ? 


Cii.  XXIII,  §§  278,  279]     PERMUTATIONS,   COMBINATIONS        871 

278.*  The  mimher  of  permutations  of  n  different  things, 
taken  r  at  a  time^  when  each  of  the  n  things  can  be  repeated^ 

is  /l^ 

After  the  first  place  has  been  filled  by  one  of  the  n 
things,  the  second  place  can  be  filled  by  any  one  of  the 
n  things  ;  and  the  first  two  places  can  be  filled  in  n^ 
ways,  etc. 

Continuing  the  process,  the  first  three  places  can  be 
filled  in  n^  ways.  The  exponent  of  n  is  evidently  the 
same  as  the  number  of  places  filled.  Hence  the  first  r 
places  can  be  filled  in  n^  different  ways.  If  x  be  the 
number  of  permutations  of  n  different  things,  taken  r 
at  a  time,  when  each  of  the  n  things  can  be  repeated, 

X  =  n\  V. 

279.*  The  number  of  permutations  of  n  things^  taken  n  at 
a  time,  when  p^  q,  and  r  -"  of  the  n  things  are  respectively 

a,  J,  and  c,  -"  is • 

p\  q\  rl  ••• 

The  proof  will  be  best  understood  by  taking  a  specific  exam- 
ple :  find  the  number  of  permutations  of  a^b^c=a  -  a-  a-b  -b-  c. 

Place  a  distinguishing  sign  of  each  of  the  three  letters  a, 
and  also  upon  the  two  letters  b,  thus :  ai,  a2,  a^,  bi,  62.  Then 
ai,  ag,  ag,  bi,  62,  c,  are  6  different  things  which  can,  by  11,  be 
arranged  in  6  !  different  ways. 

Let  X  be  the  total  number  of  permutations  of  a^b%  in  which 
3  of  the  letters  are  a,  2  are  b,  and  1  is  c.  Since,  by  II,  the  3 
letters  a,  considered  as  ai,  ag,  ag,  can  be  arranged  in  3  !  ways, 
and  the  2  letters  b  can  be  arranged  in  2 !  ways,  the  total  num- 
ber of  permutations  of  the  letters  a%%  considered  as  different 

letters,  is  a;  •  3  !  2  !,  or  6 !  =  a;  •  3 !  2 !     Hence  x  =  -^  =  60- 


372  ELEMENTARY   ALGEBKA         [Ch.  XXIII,  §  27?> 

Li  general,  let  x  represent  the  number  of  permutations 
of  n  things,  taken  n  at  a  time,  when  p^  q^  r,  •••  of  the  n 
things  are  respectively  a^  5,  c?,  •••.  If  in  any  one  of  the 
X  permutations  the  p  things  a  were  different  from  each 
other  and  all  the  others,  there  will  \}Q p  I  different  permu- 
tations instead  of  a  single  permutation.  Hence,  if  all  the 
letters  a  were  changed  into  p  different  letters,  there  would 
be  in  ^W  x  -  p\  permutations.  Similarly,  if  in  any  one  ol 
the  X'p\  permutations,  if  the  q  letters  b  were  different 
from  each  other  and  all  the  others,  there  would  hQ  x  -  p\  q^ 
permutations.  Continuing  the  process  of  changing  the 
letters  until  they  are  all  dift'erent,  the  total  number  of 
permutations  will  he  x  -  p\  q\  r  \  -".  Since  n  !  also  is 
the  total  number  of  permutations  of  n  different  letters, 
taken  ^  at  a  time,  n\=x  -  p\  q\  r\  ---^  or 

n\ 


jr  = 


p\q\ r\ 


VI. 


EXAMPLES 

1.  Find  the  numher  of  ways  in  which  a  number  of  3 
digits  can  be  formed  of  the  9  significant  digits,  repetitions 
being  allowed. 

Each  place  can  be  filled  in  9  different  ways.     Hence,  by  V, 

a:  =  9^  =z  729. 

2.  Find  the  number  of  arrangements  of  the  letters  in 
the  word  Cincinnati, 

Of  the  10  letters  in  the  word  Cincinnati,  c  is  repeated 
twice,  i  is  repeated  three  times,  and  7i  is  repeated  three  times. 

Hence,  by  VI,  .^, 

x=     ^^'     =  50,400. 
213!3! 


Cii.  XXIII,  §  279]     PERMUTATIONS  AND  COMBINATiUNS        373 

EXERCISE    CXLIV* 

1.  In  how  many  ways  can  the  following  products  be 
written  as  a  different  succession  of  factors  :  (1),  ahcdef ; 
(2),  a%c;   (3),  a%^c^ ;   (4),  a%^(^? 

2.  How  many  different  arrangements  can  be  made  of 
the  letters  in  the  following  words  :  (1),  permutation  ; 
(2),  parallel ;  (3),  combination  ;  (4),  Massachusetts;  (5),  in- 
commensurable ? 

3.  How  many  words,  of  3  letters  each,  can  be  formed 
from  a^  6,  (?,  e^  z,  (?,  u^  if  repetitions  are  allowed,  and  if  any 
order  of  letters  form  a  word  ? 

4.  How  many  numbers  of  3  digits  each,  repetitions 
being  allowed,  can  be  formed  from  the  first  5  digits  ? 

5.  How  many  odd  numbers  of  5  digits  each,  repetitions 
being  allowed,  can  be  formed  from  0,  1,  2,  •••  9  ? 

6.  How  many  even  numbers  of  4  digits  each,  repetitions 
being  allowed,  can  be  formed  from  the  digits  0,  1,  •••  9  ? 

7.  In  how  many  ways  can  groups  of  4  letters  each,  repe- 
titions being  allowed,  be  formed  from  m,  n^  r,  s,  u^  i\  w  ? 

8.  In  how  many  ways  can  groups  of  3  letters  each  be 
formed  from  the  word  Illinois  ? 

9.  In  how  many  ways  can  groups  of  3  books  each  be 
selected  from  10  books,  3  of  wliich  are  the  same  text  in 
algebra,  and  2  of  which  are  the  same  text  in  geometry  ? 

10.  How  many  different  signals  can  be  formed  from 
12  flags,  2  being  red,  3  green,  the  rest  yellow,  if  all  the 
flags,  placed  in  line,  must  be  used  to  make  a  signal  ? 


CHAPTER   XXIV 

BINOMIAL   THEOREM 

280.  The  type  forms  given  in  §  172  when  n  =  2,  3,  4, 
5,  or  6  may  be  combined  into  the  general  form 

(a  +  by  =  a^  +  na"-^b  +  "^"-^^  a/7-242 
^  ^  1-2 

n(n-\:)(n-2)^_,l^         ^^^,_ 
1.2-3 

A  proof  —  called  the  Binomial  Theorem  —  that  the  laws 
governing  the  expansion  of  (a  +  by\  when  n  is  any  posi- 
tive integer,  give  the  type  form  of  I  will  now  be  given. 

1.  That  I  is  true  when  n  =  2,  3,  4,  5,  or  6,  may  be 
seen  by  substituting  in  T,  for  example,  n  =  3. 

(a  +  5)3=  a3+  3  a%  +  3  ah'^+hK 
If  71  =  6, 

(a  +  5)6  =  ^6  +  6  a%  +  ^a^^  +  ^'^''^a%^ 
^  ^  1-2  1.2.3 

8>5.4.3^,^,     ilA:ll3l2^j5  +  6^4.3.2.1^,^ 
^1.2.3.4        ^1.2.3.4.5       ^1.2.3.4.5.6 

2.  If  I  is  true,  when  n~h^  h  being  any  positive 
integer, 

(a  +  5)^  =  a''  f  ka'-'h  +  ^^]~^^ a'^'W 

J.    •  Jj   •  O 

374 


Ch.  XXIV,  §  281]  BINOMIAL   THEOREM  375 

3.  Multiplying  both  members  of  (2)  by  a  +  5, 

1  •  A 

^  k(k-l)(k-2')^^_,^,  +  ...  +  a5* 

+  a'b  +  ka'-W'  +  ^(^-'^')  a'-^b'  +-+  Jcab"  +  5*+^ 

,  ^(^  +  1)(^)(7(;-1)^,_2^3^  ...  j^(k  +  l)ab'  +  ¥^\     (3) 

The  right  member  of  (3)  has  the  same  form  as  the  right 
member  of  (2),  (lc  +  1^  taking  the  place  of  h.  Hence  if 
the  theorem  is  true  for  any  particuhir  power,  it  is  true  for 
the  next  higher  power. 

4.  The  theorem  was  shown  in  1  to  be  true  for  the 
6th  power;  hence  it  is  true  for  the  7th  power:  being  now 
true  for  the  7th  power,  it  is  true  for  the  8th  power,  and 
so  on  for  any  power. 

5.  The  theorem  is  true  for  (^a  —  by^  since  {a  — by 
=  [«  +  (— 5)]%  the  signs  of  the  successive  terms  being 
alternate^  plus  and  minus,  the  first  term  being  plus. 

281.  Any  required  term  can  be  written  without  com- 
pleting the  expansion  by  observing  the  laws  for  the  for- 
mation of  particular  terms.  Thus,  the  fourth  term  of 
(^a  +  by  is  known  to  be  ^0^-- ^X^- 2)^^.3^3^  ^j^^  ^j^.^^ 

A.  *  A  '  O 

term  of   (a-\-by+^  is  known  to   be    ^^  +  ^^^a^-'6^  etc. 

1  •  2 


87G  ELEMENTARY   ALGEBRA       [Cii.  XXIV,  §  28'2 

Similarly  the  rtli  term  of  (a  +  5)"^  is, 

n(n  -V)(n-  2)  «»»  (n-r-\-  3X^  -  r  +  2) ^^-r+ijr-i . 
1.2.8...  (r-2)(r-l) 

and  the  (r  +  l)st  term  of  (a  +  J)^  is, 

n(n  —  D(n  —  2^  ■-  Qi  —  r  +  2)(7^  —  r  +  1) ^T^-rjr 
1-2-8  •••  {r  —  l){r) 

282.     The  number  of  terms  in  the  expansion  of  (a  +  5)% 
when  71  is  a  positive  integer,  is  limited.     Thus,  by  I, 

(a  +  J)4  =  a^  +  4  aSJ  +  Il|a2t2  +  i_L?JL|^J3 

i  .  Z  1  •  ^  •  o 

.  4.3. 2. K,  ,  4.3.2.1.0   _i.5 


1.2.3-4         1.2.3.4.5 

Since  the  coefficients  of  all  terms  following  the  fifth 
contain  a  zero  factor,  all  such  terms  disappear.  In  general, 
if  n  is  a  positive  integer,  the  expansion  of  (a  +  by  ends 
with  the  (n  +  l)st  term. 

The  coefficients  of  terms  equally  distant  from  the  end 
terms  are  equal.     It  is  evident  that 

(^a  +  hy=(b  +  ay. 

(h  +  ay  =  h''  +  7ih''-^a^'^^^^^^b--'^(^-  +  -.  nJa'^-l  +  a^    (4) 

(b  +  ay  is  merely  the  expansion  ot  (a  +  by  written  in  de- 
scending powers  of  6.  The  last  term  of  I  is  the  same  as 
the  first  term  of  (4)  ;  the  second  term  of  I  is  the  second 
from  the  last  of  (4),  etc. 

Hence  in  the  expansion  of  a  binomial,  terms  after  the 
middle  term  Qor  teryns)  take  their  coefficients  in  reverse  order. 


Ch.  XXIV,  §  282]  BINOMIAL   THEOREM  377 

EXAMPLES 

1.    Expand  (3  a  -  1/. 
By  I, 

(3a-l/=(3a)^  +  5(3ay(-l)+f^(3a)X-l)2 

5.4.3.2.1,  ,,5 


:(-!/ 


1.2.3.4.6' 
=  243a'-405a<+270a8-90a2  +  i5a-l, 

2.    Find  the  first  4  terms  and  the  last  4  terms  of  (x  —  y)3i. 
By  I,  and  §  282, 

. ■  31 .  30  .  29  ,^,,,     ..,2«  ,  31  .  30,..,,,      .^ 


'.{xYi-yy^^-'-^l^ixfi-yf 


1-2.3     ^  '  ^     "'  1-2 

+  31(x)(-s,)«'  +  (-y)3i 
=  a;''  -  31  a^y  +  465  a;^^^  -  4495  a^/  ... 

+  4495  ar"^^  -  465  x'y"^  +  31  mj^  -  f\ 

8.    Find  the  6th  term  of     (l-^^^. 
By  §  281,  the  6th  term  of 

3  .  7  .  11  5V6 


2< 

231  6^V6 
16 


378  ELEMENTARY   ALGEBRA      [Ch.  XXIV,  §  281! 

EXEECISB  CXLV 

Expand  the  following  binomials : 

2.  (a -52)5.  8.   (2x-yy.  '   ^*      ^^^ 

3.  (a2+52y.  9_  (a-2xy.  13.  (\--l)*- 

4.  (l  +  x2y.  10.   (3rr-2^)5.  14.   (2Va-l)6. 

5.  CaJ  — 1)".  /I  \6  /      1      1\6 

6.  (a;-i  + 1/-2)5,  Vx  V  \            bj 

16    /2a-i     aV^Y  19    ('^a-i     aV^V 

„    /2«VJ       lY  20    /^2aVF3      Viy\s 

/9  rt        -r-v 

18 


'•  \-^-^'  \     b     .      3V3^ 

Express  in  simplest  form  the  indicated  terms  of  the 
following  binomials : 

22.  4th  term  of  {x  —  yy.    23.  2d  term  of  {x  —  yy'\ 

24.  11th  term  of  (a  -  by\ 

(  3  5"2\31 

25.  5th  term  of  (  o^b 


B  b-^ 

26.  6th  term  of  f-^  -  ^^y. 

\7  bVb     VSaJ 

27.  8thtermo£f^--Ji^^. 

\  b        la) 

28.  10th  term  of  {—  -  — Y^- 


Ch.  XXIV,  §  282]  BINOMIAL   THEOREM  379 

2V^     6a/P\21 


29.  6th  terra  of  i  — — — i  • 

\   6  a    J 

30.  8tli  and  11th  terms  of  f^^  -  6 VpY^ 

31.  4th  and  17th  terms  of  {4^ ^^T* 


f         2  \* 
33.   (n  —  2)d  term  of    a ) 


32.   (r  +  l)st  term  of  (2  a  -  hy 

2  \*+^ 

34.  Find  the  first  4  and  the  last  4  terms  of  ( Va  -  2-^)20. 

35.  Find  the  first  6  and  the  last  3  terms  of  (l  -  ^V^Vl 

36.  Find   the   terms  that  do   not   contain   radicals   in 


37.  Find  the  coefficient  of  x^^  in  (a;  +  2  x^y^. 

38.  Find  the  coefficient  of  a^  in  [  a  +  -  j   • 

39.  Find  the  coefficient  of  a^^  in  (  a? )   • 

40.  Find  the  term  independent  of  h  in  [\\  —  ^^ 

41.  Find  the  term  independent  of  x  in  ( \^  —  ^J   • 

(2a        :z:  \' 
— _ _j 


CHAPTER  XXV 

LOGARITHMS 

283.  The  logarithm  of  any  number  is  the  exponent  indi- 
cating the  power  to  which  a  certain  fixed  number,  called 
the  base,  must  be  raised  in  order  to  produce  the  given 
number. 

EXAMPLES 

1.  Find  the  logarithm  of  25  if  the  base  is  5. 
Since  26  =  (5)^,  the  logarithm  of  25  is  2. 

2.  Find  the  logarithm  of  243  if  the  base  is  9. 

Since  243= (3/=  (32)^= (9)^,  the  logarithm  of  243  is  |==2.5. 

3.  Find  the  logarithm  of  16  if  the  base  is  8. 

Since  16  =  (2)^  =  (23)t  =  (8)*,  the  logarithm  of  16  is 
1  =  1.3333.-.. 

4.  Find  the  logarithm  of  ^j  if  the  base  is  3. 

Since  —  =  -—  =  (3)"^,  the  logarithm  of  -^^  is  —  3. 

27      (3) 

EXERCISE  CXLVI 

Find  the  logarithms  of  the  following  numbers :  « 

1.  8,  32,  2  V2,  |,  yl^,  the  base  being  4. 

2.  3,  27,  81 V3,  1  ^\,  the  base  being  9. 

3.  2,  \,  232,  _J_.^  the  base  being  16. 

380 


Ch.  XXV,  §§  284-286]  LOGARITHMS  381 

284.  In  the  common  (or  Briggs)  System,  the  number  10 
is  always  taken  as  the  base.     It  may  be  shown  that 

100=1,  10<>  =  1, 

10-100,  io-  =  i^,=  o.oi, 

103  =  1000,  10-^=105-0-001, 

10^  =  10000.  10-^  =  ^=0.0001. 

285.  Log  1  =  0  is  a  short  way  of  writing  that,  in  the 
system  in  which  the  base  is  10,  the  exponent  of  the  power 
of  10,  which  produces  1,  is  0.     Hence, 

log  1  =  0,  log  1  =  0, 

log  10  =  1,  log  0.1  =  -1, 

log  100  =  2,  log  0.01  =  -2, 

log  1000  =  3,  log  0.001  =  -  3, 

log  10000  =  4.  log  0. 0001  =  -  4. 

286.  It  is  evident  that  a  number  between  1  and  10  has 
a  logarithm  between  0  and  1  ;  a  number  between  10  and 
100  has  a  logarithm  between  1  and  2  ;  a  number  between 
100  and  1000  has  a  logarithm  between  2  and  3 ;  a  number 
between  1  and  0.1  has  a  logarithm  between  0  and  —1  ; 
a  number  between  0.1  and  0.01  has  a  logarithm  between 

1  and  —2;  a  number  between  0.01  and  0.001  has 
a  logarithm  between  —2  and  —3,  etc.  In  general,  the 
logarithm  of  a  number  greater  than  1  is  positive,  and 
the  logarithm  of  a  number  less  than  1  is  negative. 


382  ELEMENTARY  ALGEBRA        [Ch.  XXV,  §  287 

287.  The  logarithm  of  a  number,  not  an  exact  power 
of  10,  consists  of  two  parts,  —  the  characteristic,  which  is 
the  integral  part,  and  the  mantissa,  which  is  a  fractional 
part  expressed  as  a  decimal. 

The  characteristic  of  the  logarithm  of  any  number 
greater  than  1  is  always  positive,  and  depends  upon  the 
number  of  significant  digits  in  the  number  to  the  left  of 
the  decimal  point.  From  the  table  in  the  preceding  para- 
graph, it  may  be  seen  that  any  number  containing  two 
digits  to  the  left  of  the  decimal  point  has  a  characteristic 
of  1  ;  that  any  number  containing  three  digits  to  the  left 
of  the  decimal  point  has  a  characteristic  of  2,  etc.    Hence  : 

The  characteristic  of  the  logarithm  of  any  number  greater 
than  1  is  always  one  less  than  the  number  of  digits  preceding 
the  decimal  point. 

The  characteristic  of  the  logarithm  of  any  number  less 
than  1  is  always  negative,  and  depends  upon  the  number 
of  zeros  between  the  decimal  point  and  the  first  signifi- 
cant digit.  From  the  table  in  the  preceding  paragraph, 
it  may  be  seen  that  any  number  less  than  1  and  contain- 
ing no  zeros  between  the  decimal  point  and  the  first 
significant  digit  is  —  1  ;  that  any  number  containing  one 
zero  between  the  decimal  point  and  the  first  significant 
digit  is  —  2,  etc.  The  characteristic  of  the  logarithm 
of  a  number  less  than  1  is  rarely  written  in  a  negative 
form,  but  thus : 

—  1  is  written  9(+  decimal)  —  10, 

—  2  is  written  8(4-  decimal)  — 10, 

—  3  is  written  7(+  decimal)  —  10. 


Ch.  XXV,  §§  288, 289]  LOGARITHMS  383 

The  logarithm  of  a  number  less  than  1  will  have  a 
characteristic  which  is  the  difference  between  9  and  the 
number  of  zeros  between  the  decimal  point  and  the  first 
significant  digit,  minus  10.     Hence  : 

The  characteristic  of  the  logarithm  of  any  number  less 
titan  1  is  negative^  and  is  the  difference  between  9  and  the 
number  of  zeros  between  the  decimal  point  and  the  first  sig- 
nificant digits  writing  —  10  after  the  mantissa, 

288.  The  mantissa  of  the  logarithm  of  any  number  is 
given  in  the  table  on  pages  394  and  395. 

PRINCIPLES   OF   LOGARITHMS 

289.  I.  The  logarithm  of  the  product  of  two  or  more  fac- 
tors is  the  sum  of  the  logarithms  of  the  factors. 

Let  10"^  =  X,    or    log  x   =  a^  (1) 

and  let  10^  =  ?/,    or    log  «/  =  5,  (2) 

multiplying  (1)  and  (2), 

IQci+b  __  ^y^  Qp  i^g  xg  =  a  +  b  =  log  x  +  log  y.  (3) 

Similarly,  I  can  be  proved  for  the  product  of  three  or 
more  factors. 

II.  The  logarithm  of  the  quotient  of  two  numbers  is  the 
logarithm  of  the  dividend  minus  the  logarithm  of  the  divisor. 

Let  lO"*  =  x^    or    log  x  =  a,  (1) 

and  let  10*  =  y,    or    log  «/  =  J,  (2) 

dividing  (1)  by  (2), 

10«  -  ^  =  -,  or  log  -  =  a  -  5  =  log  :r  -  log  y.        (3) 

i/  y 


384  ELEMENTARY   ALGEBRA         [Ch.  XXV,  §  290 

III.  The  logarithm  of  the  power  of  a  number  is  the  prod- 
uct of  the  logarithm  of  the  number  by  the  exponent  of  the 
poiver. 

Let         10"  =  x^    or  log  a;  =  a,  (1) 

raising  both  members  of  (1)  to  the  5th  power, 

10"^  =  x^,  or  log  x^=^ab^b  log  x,  (2) 

IV.  The  logarithm  of  the  root  of  a  number  is  the  quotient 
obtained  by  dividing  the  logarithm  of  the  number  by  the  index 
of  the  root. 

Let         10"*  =  x^    or  log  x  =a^  ,  (1) 

extracting  the  5th  root  of  both  members  of  (1), 

m=x\  or  logx^  =  j  =  ^-^  =  \]ogx.        (2) 
b  b  b 

Note.     The  above  principles  hold  for  any  number  whatever. 

290.  The  mantissa  of  the  logarithms  of  all  numbers  which 
have  the  same  sequence  of  digits  is  the  same. 

Let        log  214.5  =  2.3314, 

then  log  2145  =  log(214.5x  10)  =log  214.5-f  log  10 

=  2.3314  +  1  =  3.3334. 

Let        log  214.6  =  2.3314, 

then     log  0.002145  =  log(214.5  --  100,000) 

=  log  214.5 -log  100,000 

=  2.3314  -  5  =  7.3314  -  10. 

From  the  above  examples,  it  is  evident  that  changing 
the  position  of  the  decimal  point  is  merely  multiplying  or 
dividing  the  given  number  by  a  power  of  10. 


Ch.  XXV,  §§  291, 292]  LOGARITHMS  385 

USE   OF   THE  TABLE 

291.  To  find  the  logarithm  of  a  number  consisting  of 
three  digits : 

On  pages  394-395  find  in  the  column  under  N  the  first 
two  digits  of  the  given  number.  The  mantissa  required  will 
he  foxmd  at  the  intersection  of  the  horizontal  line  containing 
the  first  tivo  digits  and  the  vertical  column  headed  by  the 
third  digit.     Prefix  the  proper  characteristic, 

log  21.7  =  1.3365, 

log  0,429  =  9.6325 -10, 

log  970  =  2.9868, 

log  0.0211  =  8.3243 -10. 

Numbers  containing  less  than  three  digits  are  similarly 

^^^^^-  log  0.27  =  9.4314  - 10, 

log  5  =  0.6990, 

log  0.0029  =  7.4624  - 10. 

292.  To  find  the  logarithm  of  a  number  consisting  of 
more  than  three  digits: 

1.    Find  the  logarithm  of  92.04. 

Mantissa  of  the  log  of  the  sequence  920  =  9638, 

mantissa  of  the  log  of  the  sequence  921  =  9643. 

An  increase  of  one  unit  in  the  sequence  gives  an  increase 
of  0.0005  in  the  mantissa;  an  increase  of  0.4  of  a  unit  in  the 
sequence  gives  an  increase  of  0.4  x  0.0005  =  0.0002  in  the 
mantissa.     Therefore,  * 

mantissa  of  the  log  of  the  sequence  9204  =  9640, 

prefixing  required  characteristic,  log  92.04  =  1.9640. 


386  ELEMENTARY  ALGEBRA        [Ch.  XXV,  §  293 

2.    Find  the  logarithm  of  0.01238. 

Mantissa  of  the  log  of  the  sequence  123  =  0899, 

mantissa  of  the  log  of  the  sequence  124  ~  0934. 

An  increase  of  one  unit  in  the  sequence  gives  an  increase 
of  0.0035  in  the  mantissa;  an  increase  of  0.8  of  a  unit  in  the 
sequence  gives  an  increase  of  0.8  x  0.0035  =  0.0028  in  the  man- 
tissa.    Therefore 

mantissa  of  the  log  of  the  sequence  1238  =  0927, 

prefixing  required  characteristic,  log  0.01238  =  8.0927  — 10. 

293.  The  process  of  making  the  proper  correction  in 
the  logarithms  of  numbers  of  more  than  three  digits  is 
called  Interpolation,  and  is  based  upon  the  hypothesis  that 
adjacent  mantissas  increase  proportionally  with  the  corre- 
sponding numbers.  Corrections  made  in  this  manner  are 
not  strictly  accurate ;  and  even  the  mantissas  given  are 
only  approximate,  but  are  correct  to  0.00005.  If  the  cor- 
rection in  the  fifth  decimal  place  be  5  or  more,  the  fourth 
decimal  place  is  increased  by  1. 

In  the  table  on  pages  394-395  find  the  mantissa  of  the 
first  three  significant  digits^  disregarding  the  position  of  the 
decimal  point;  subtract  the  mantissa  thus  found  from 
the  mantissa  of  the  next  higher  number  of  three  significant 
digits;  multiply  the  difference  thus  found  by  the  decimal 
represented  by  the  remaining  digits  of  the  given  number; 
add  the  product  (to  the  fourth  decimal^  to  the  mantissa 
of  the  first  three  digits.     Prefix  the  proper  characteristic. 


Ch.  XXV,  §  294]  LOGARITHMS  387 

294.  To  find  the  number  corresponding  to  a  given 
logarithm. 

1.    Find  the  number  whose  logarithm  is  7.5521  — 10. 

From  the  table,  5514  is  the  mantissa  of  the  sequence  356, 
and  5527  is  the  mantissa  of  the  sequence  357 ;  that  is,  a  dif- 
ference of  0.0013  in  the  mantissa  gives  a  difference  of  one  unit 
in  the  sequence ;  hence  the  mantissa  5521,  being  0.0007  more 
than  the  mantissa  5514,  gives  a  difference  of  -^^  of  one  unit 
(=0.5)  in  the  sequence.     Therefore,  applying  §  287, 

log  0.003565  =  7.5521  - 10. 

The  number  corresponding  to  a  given  logarithm  is  called 
the  anii/ogarithm. 

EXERCISE   CXLVII 

Find  the  logarithms  of  the  following  numbers : 


1. 

254. 

7. 

362. 

13. 

8.437. 

2. 

465. 

8. 

5685. 

14. 

0.003. 

3. 

200. 

9. 

6297. 

15. 

0.000569. 

4. 

908. 

10. 

1004. 

16. 

0.009186. 

5. 

2. 

11. 

0.8562. 

17. 

0.01089. 

6. 

20. 

12. 

0.008547. 

18. 

0.9989. 

Find  the 

antiloga 

rithms  of  : 

19. 

0.3927. 

25. 

0.9821. 

31. 

0.0250. 

20. 

1.6395. 

26. 

1.6872. 

32. 

9.5299-10. 

21. 

8.7235. 

27. 

3.5689. 

33. 

8.7467-10. 

22. 

9.8420- 

-10. 

28. 

5.6372. 

34. 

2.8837. 

23. 

7.9069- 

-10. 

29. 

4.3204. 

35. 

8.9432-10. 

24. 

6.9903- 

-10. 

30. 

2.3974. 

36. 

7.0161-10. 

388  ELEMENTARY   ALGEBRA         [Ch.  XXV,  §  291^ 

USE   OF   LOGARITHMS   WHICH   HAVE   NEGATIVE 
CHARACTERISTICS 

295.    In  finding  the  antilogarithm  of<  a  negative  logarithm^ 
— 10  should  always  appear  at  the  end  of  the  logarithm. 

EXAMPLES 
1.   Add  the  following  logarithms : 

9.6253  - 10 
8.5145-10 


18.1398  -  20  =  8.1398  - 10. 

2.    Subtract  the  logarithm  3.1461  from  the  logarithm 

9  14^0 

^'  ■*'^^'         2.1430  =  12.1430  - 10 

3.1461=  3.1461 


8.9969  - 10. 

3.    Subtract  the  logarithm  9.3141  — 10  from  the  loga- 
rithm 8.6537-10. 

8.6537-10  =  18.6537-20 
9.3141-10=   9.3141-10 


9.3396-10. 


4.   Multiply  the  logarithm  8.1461  -  10  by  2. 

8.1461  - 10 

2 

16.2922-20  =  6.2922  - 10. 

6.    Divide  the  logarithm  7.9101  -  10  by  3. 

7.9101  - 10  =  27.9101  -  30 

3)27.9101  -  30 
9.3034  - 10, 


Cii.  XXV,  §  295]  LOGARITHMS  388 

In  multiplying  a  logarithm  hy  a  fraction^  multiply  the 
logarithm  hy  the  numerator  and  divide  this  product  by  the 
denominator,  in  the  order  stated,  taking  care  to  simplify  at 
each  step. 

6.   Multiply  the  logarithm  8.3196  -  10  by  f . 

8.3196  - 10 

2 


16.6392  -  20  =  26.6392  -  30 

3)26.6392  -  30 
8.8797  - 10. 


EXERCISE   CXLVIII 

Perform  the  indicated  operations  in  the  following  loga- 
rithms : 

1.  (9.7305 -10) +  (9.3457 -10). 

2.  (8.5478 -10) +  (9.8438 -10). 

3.  (0.6544) +  (9. 7258 -10). 

4.  (0.8733) -(2. 7459). 

5.  (9.3476) -(9.5244).  8.   (9.1436-10)  x  4. 

6.  (8.2386  - 10)  X  5.  9.    (6.8433 -10)  x|. 

7.  (8.8300 -10)-!- 3.  10.    (9.8010- 10) -i-|. 

11.  (7.1431- 10)  xf  +  (8.7153- 10). 

12.  (2.5157)  xi- (9.9918- 10). 

13.  (6.5000)  -  (8.5431)  x  |. 

14.  (7. 2511  - 10)  +  (8.2190)  x  f . 

15.  (9.0909)  X  5  -  (8.1650)  x  |. 

16.  (2.0001)  X  f  -(8.0999)  x  f 


39G  ELEMENTARY  ALGEBRA         [Ch.  XXV,  §290 

COMPUTATIONS  BY   LOGARITHMS 

ooc    i     T7-   ^  4-1.        1        f  192.7  X  6.54  X  0.4683 
296.    1.    Find  the  value  of  ^^^^^^^^^^^^^-^. 

log  192.7  =  2.2849  log  1624  =  3.2106 

log  6.54  =  0.8156  log  0.0329  =  8.5172-10 

log  0.4683  =  9.6705-10  log  1 .028  =  0.0120 

log  numerator  =  2.7710  log  denominator  =  1.7398 

log  denominator  =  1.7398 
log  fraction  =  1.0312 
fraction  =  10.75 

2.  Find  the  value  of  V32.5  x  68.7  x  32.74. 

log  32.5  =  1.5119 

log  68.7  =  1.8370 

log  32.74  =  1.5151 

log  product  =  4.8640 

^  log  product  =  2.4320 

product  =  270.4. 

3.  Findthe  value  of  (5.235)3. 

log  5.235  =  0.7189 

3  log  5.235  =  2.1567 

(5.235)3  =  143.5. 

4.  Find  the  value  of  0.763  x  62.8  +  8632  -^  3.265. 

log  0.763  =  9.8825  - 10  log  8632  =  3.9361 

log  62.8  =  1.7980  log  3.265  =  0.5139 

log  product  =  1.6805  log  quotient  =  3.4222 

product  =     47.92  quotient  =  2644. 
quotient  =  2644. 
sum  =  2691.92. 

Note.    The  last  two  digits  are  not  accurate  since  a  four-place 
table  is  used- 


Ch.  XXV,  §  296]  LOGARITHMS  391 

5.  Find  the  value  of  —  V8  x  -^^. 

log  8  =  0.9031  log  1  =  10.0000 -10 

■^  log  8  =  0.4516  log  7  =   0.8451 

-I  log  ^  =  9.7183  - 10  log  I  =  29.1549  -  30 

log  product  =  0.1699  i  log  j-  =   9.7183  - 10 
product  =  -  0.1479. 

Note  that  the  product  is  negative  in  accordance  with  the  law 
of  signs. 

6.  Solve  the  equation  3-^  =  4,  by  the  use  of  logarithms. 

log  3^  =  log  4, 
a;  log  3  =  log  4, 

log  3     0.4771 

Notice  that  the  above  example  is  a  case  of  an  irrational 
number  employed  as  exponent. 

EXERCISE   CXLIX 

Compute  by  the  use  of  logarithms  : 

1.  21.4x9.87.  11.  251.2 --0,785. 

2.  6.92x58.4.  12.  0.09891  H- 0.001234. 

3.  0.908x201.  13.  200.9-^10.01. 

4.  65.31x0.319.  14.  8957^0.9081. 

5.  0.8642x589.7.  is.  0.7154 -i- 9.003. 

6.  0.9034x0.00154.  16.  0.2167  h- .0.0375. 

7.  698-5-20,  17.  0.04678-^892. 

8.  0.583 -f- 2982.  la  0.0001-5-894.5. 

9.  0.9085-1-9.805.  19,  8.9x0.32x0.065. 
10.  0.9651-1-0.8939.  20.  0.8x3x500. 


392  ELEMENTARY  ALGEBRA  [Ch.  XXV,  §296 

21.  0.3  X  0.09  X  0.1986.  „„  6456  x  0.6456  x  0.06456 

22.  6.98x0.6851x0.32.  27x270x2700 

23.  0.91x0.81x0.09.  29  0-4692  x  9231  x  64.82 

24.  0.0061x3159^0.005468.  '  0.1492  x  0.8361  x  6987" 
„  6.83x0.7816x0.9181  30.  0.5533x419.2x0.3265. 
^^-        9.2184  X  0.07436      '  60.90x5.432x0.8406 

„    215.4  X  89.72  x  0.896       31    6384  x  0.0987  x  0.012 
0.6671x19.2x88.32'         '   2007x0.3388x0.871 

„„    2.754x0.9803x2001       ,^    0.7188x0.8159x0.0001 
*    3721x0.1596x0.31  '0.01897x0.8963x0.3031* 

33.    (6.608)2.  39.    V6479T.  45.    -v^O.OOOS. 


34.  (2.755)2.  40.   V9381.  46.    -v/0.2756- 

35.  (1.01)25.  41.   ^0.0182.  47.    •v'0.1622- 

36.  (99.81)3.  42.    ^6503.  48.    A/85r2. 

37.  (49.73)*.  43.    ■v'50.  49.    ^f. 

38.  (0.9801)6.  44.    ^0.1257.  50.    -v/|. 

I  23  X  75  „      3/0.152  X  0.025 

51.    -VI •  56.    ■\/ • 

>'l3x0.85  ^    25x0.085 


y^. 


g2       ,J.525  X  0.054  gy    j/0.3756  x  0.265 


351  X  0.062  *    ^    0.227  x  863 

JO^ 


„       10.768  X  0.0345  gg    (0.03472)^  x  -v/4011 
2512.x  0.071  '  ■  (1.21) 

,,    J2. 01-6  X  0.06932  „    5076  VO.  007109 

54.    \/ •  3"»    „  — • 

^    0.1126x987  9834-V/0.045 

3|0.0435  X  3986  go        (0.3143)^     ^ 
'   ^' 4534  X  0.087  '  ■l.63-V0.163 


Ch.  XXV,  §  296J  LOGARITHMS  393 


61.  (|)3v/36.  64.    Vs+^T.  67.    ■j^jV'Jf. 

62.  •v'o.aSVa.  65.    ^384  +  ^81.     68.    -^mWM. 

63.  (11)6^8721.      66.    -^^1  -</2i.     69.    (i|)*v'0:8557 


,^_   ^43  +  5£278.  ^^_   ^_A19M^. 

-^17  8097V0.85 


71.   5Vif-V0.674.  ^g    ^6.923-|</9999 


72.    V2.7  +  3\/0.15. 

73 


V0.1807 


^ 


52.38-17^^088 


V9.921- 3^502.  -^-    ^— ^T^^^ 

74.    ^0.783 -6V0.0431.         78.    [(1.048)3 --v/OlT]*. 

79.  Solve  for  a: :  3^  =  13. 

80.  Solve  for  a;:  12^  =  25. 

81.  Solve  for  a;:  6^=54.83. 

82.  Solve  for  a;:  3-^=1.923. 

83.  Solve  for  rr:  5^=0.1987. 

84.  Solve  for  2;:  2a^=1.62b4:. 

85.  Solve  for  x:  (2  +  0.3)^=  10. 


394 


ELEMENTARY   ALGEBRA         [Cit  XXV,  §  296 


N 

0 

1 

2 

3 

4 

6 

6 

7 

8 

9 

10 
11 
12 

0000 
0414 
0792 

0043 
0453 
0828 

0086 
0492 
0864 

0128 
0531 
0899 

0170 
0569 
0934 

0212 
0607 
0969 

0253 
0645 
1004 

0294 
0682 
1038 

0334 
0719 
1072 

0374 
0755 
1106 

13 
14 
15 

1139 
14H1 
1761 

1173 
1492 
1790 

1206 
1523 
1818 

1239 
1553 
1847 

1271 

1584 
1875 

1303 
1614 
1903 

1335 
1644 
1931 

1367 
1673 
1959 

1399 
1703 
1987 

1430 
1732 
2014 

16 
17 
18 

2041 
2304 
2553 

2068 

23;ho 

2577 

2095 
2:i55 
2601 

2122 

2;^ 

2625 

2148 
2405 
2648 

2175 
2430 
2672 

2201 
2455 
2695 

2227 
2480 
2718 

2253 
2504 
2742 

2279 
2529 
2765 

19 
20 
21 

2788 
3010 
3222 

2810 
3032 
3243 

2833 
3054 
3263 

2856 
3075 
3284 

2878 
:3096 
3304 

2900 
3118 
3324 

2923 
3139 
3345 

2945 
3160 
3365 

2967 
3181 
3385 

2989 
3201 
3404 

22 
23 
24 

3424 

3617 
3802 

3444 
3636 
3820 

3464 
3(555 
3838 

3483 
3674 
3856 

3502 
3692 
3874 

3522 
3711 
3892 

3541 
3729 
3909 

3560 
3747 
3927 

3579 
376(5 
3945 

3598 
3784 
3962 

25 
26 

27 

3979 
4150 
4314 

3997 
4166 
4330 

4014 
4183 
4346 

4031 
4200 
4362 

4048 
4216 
4378 

4065 
4232 
4393 

4082 
4249 
4409 

4099 
4265 
4425 

4116 
4281 
4440 

4133 
4298 
4456 

28 
29 
30 

4472 
4<)24 
4771 

4487 
4639 
4786 

4502 
4(554 

4800 

4518 
4669 
4814 

4533 
4683 
482<) 

4548 
4698 
4843 

4564 
4713 

4857 

4579 

4728 
4871 

4594 
4742 

4886 

4609 
4757 

31 
32 
33 

4914 
5051 
5185 

4928 
5065 
5198 

4942 
5079 
5211 

4955 
5092 
5224 

4969 
5105 
5237 

4983 
5119 
5250 

4997 
5132 
5263 

5011 
5145 
5276 

5024 
5159 
5289 

5038 
5172 
5302 

34 
35 
36 

5315 
5441 
5563 

5328 
5453 
5575 

5310 
5465 
5587 

5353 
5478 
5599 

536(5 
5490 
5611 

5378 
5502 
5623 

5391 
5514 
5635 

5403 
5527 
5647 

5416 
5539 
5658 

5428 
5551 
5670 

37 
38 
39 

5682 
5798 
5911 

5694 
5809 
5922 

5705 
5821 
5933 

5717 
5832 
5f)44 

5729 
5843 
5955 

5740 

5855 
5966 

5752 

58(56 
5977 

5763 

5877 
5988 

5775 
5888 
5999 

5786 
589<) 
(5010 

40 
41 
42 

6021 
6128 
6232 

6(J31 
6138 
6243 

6042 
6149 
6253 

6053 
6160 
6263 

6064 
6170 
6274 

(5075 
6180 
6284 

6085 
6191 
6294 

609(5 
6201 
6304 

6107 
6212 
6314 

6117 
6222 
6325 

43 
44 
45 

6335 
6435 
6532 

6345 
6444 
6542 

6355 
6454 
6551 

6365 
(5464 
6561 

6375 
6474 
6571 

6385 
(5484 
6580 

6395 
(5493 
6590 

6405 
6503 
6599 

6415 
6513 
6609 

6425 
6522 
6618 

46 

47 
48 

()()28 
()721 
(3812 

6637 

67;^ 

6821 

61^46 
()73<) 
6830 

mm 

6749 
6839 

6665 
6758 
6848 

6675 
()7()7 
6857 

6684 
6776 
6866 

6693 
6785 
6875 

6702 
6794 

6884 

6712 
6803 
6my 

49 
50 
51 

6fK)2 
69<)0 
7076 

6911 
6998 
7084 

6920 
7007 
7093 

6f)28 
7016 
7101 

6937 
7024 
7110 

6946 
7033 
7118 

6955 
7042 
7126 

6964 
70.50 
7135 

6972 
7059 
7143 

6981 
70(57 
7152 

52 
53 
54 

7160 
7243 
7324 

7168 
7251 
7332 

7177 
7259 
7340 

7185 
7267 
7348 

7193 

7275 
7356 

7202 
7284 
7364 

7210 
7292 

7372 

7218 
7300 
7380 

7226 

7308 
7388 

7235 
7316 
7396 

Cn.  XXV,  §  206j 


LOGARITHMS 


395 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

55 
56 
57 

7404 

7482 
7559 

7412 
7490 
7566 

7419 
7497 
7574 

7427 

7505 
7582 

7435 
7513 
7589 

7443 
7520 
7597 

7451 

7528 
7604 

7459 
7536 
7612 

7466 
7543 
7619 

7474 
7551 
7627 

58 
59 
60 

7634 
7709 

7782 

7642 
7716 
7789 

7649 
7723 
7796 

7657 
7731 
7803 

7664 
7738 
7810 

7672 
7745 
7818 

7679 

7752 
7825 

7686 
7760 
7832 

7694 

77(>7 
7839 

7701 

7774 
7846 

61 
62 
63 

7853 
7924 
7993 

7860 
7931 
8000 

7868 
7938 
8007 

7875 
8014 

7882 
7952 
8021 

7889 
7959 
8028 

7896 
7^)66 
8035 

7903 
7973 
8041 

7910 

7980 
8048 

7917 

7987 
8055 

64 
65 
66 

8002 
8129 
8195 

8069 
8136 
8202 

8075 
8142 
8209 

8082 
8149 
8215 

8089 
8156 
8222 

80V)6 

8162 
8228 

8102 
8169 
8235 

8109 
817() 
8241 

8116 

8182 
8248 

8122 
8189 
8254 

67 
68 
69 

8261 
8325 
8388 

8267 
8331 
8395 

8274 
8338 
8401 

8280 
8:m 
8407 

8287 
8351 
8414 

8293 
8357 
8420 

8299 
8363 
8426 

8306 
8370 
8432 

8312 
8376 
8439 

8319 
8382 
8445 

70 

71 
72 

8451 
8513 
8573 

8457 
8519 
8579 

8463 
8525 
8585 

8470 
8531 
8591 

8476 
85.'57 
8597 

8482 
8543 
8()03 

8488 
8549 
8609 

8494 
8555 
^15 

8500 
8561 
8621 

8506 
8567 
8627 

73 

74 
75 

8()33 
8()92 
8751 

8639 
8698 
8756 

8()45 
8704 
8762 

8651 
8710 
8768 

8657 
8716 
8774 

86()3 
8722 
8779 

86()9 

8727 
8785 

8675 
8733 
8791 

8681 
8739 
8797 

8686 
8745 
8802 

76 

77 
78 

8808 
8865 
8921 

8814 
8871 
8927 

8820 
8876 
8932 

8825 
8882 
8938 

8831 
8887 
8943 

8837 
8893 
8949 

8842 
8899 
8954 

8848 
8^)04 
8960 

8854 
8910 
8965 

8859 
8915 
8971 

79 
80 
81 

8970 
<K)31 
9085 

8982 
903(; 
9090 

8987 
9042 
909(i 

8993 
9047 
9101 

8998 
9053 
9106 

9004 
9058 
9112 

9009 
9063 
9117 

9015 
90()9 
9122 

9020 
<)074 
9128 

9025 
9079 
9133 

82 
83 
84 

9138 
9191 
9243 

9143 
91(K> 
9248 

9149 
9201 
9253 

9154 
9206 
9258 

9159 
9212 
9263 

9165 
9217 
9269 

9170 
9222 
9274 

9175 
9227 
9279 

9180 
9232 

9284 

9186 
9238 
9289 

85 
86 
87 

9294 
9;M5 
9395 

9299 
9350 
9400 

91304 
9355 
9405 

9309 
93()0 
9410 

9315 
93(J5 
9415 

9320 
9370 
9420 

9325 
9375 
9425 

9330 
9380 
9430 

9335 
9385 
9435 

9340 
93<X) 
9440 

88 
89 
90 

9445 
9542 

9450 
9499 
9547 

9455 
9504 
9552 

94()0 
9509 
9557 

9465 
9513 
9562 

9469 
9518 
95()6 

9474 
9523 
9571 

9479 
9528 
9576 

9484  . 

9533 

9581 

9489 
9538 
9586 

91 
92 
93 

9590 

-  9638;. 

9685 

9595 
9()4a 
9689 

9600 
9(i47 
9694 

9605 
9652 
9699 

9609 
9(557 
9703 

9614 
9()61 
9708 

9619 
9(56() 
9713 

9()24 
iK)71 
9717 

9()28 
%75 
9722 

9()33 
9680 
9727 

94 
95 
96 

9731 
9777 
9823 

9736 

W82 
9827 

9741 

9786 
9832 

9745 
9791 
9836 

9750 
9795 
9841 

9754 

9800 
9845 

9759 

9805 
9850 

9763 
9809 
9854 

9768 
9814 
9859 

9773 

9818 
9863 

97 
98 
99 

9868 
9912 
9956 

9872 
9917 
9961 

9877 
9^)21 
9965 

9881 
9926 
9969 

9886 
9930 
9974 

9890 
9934 
9978 

9894 
9939 
9^)83 

9899 
9943 
9987 

9903 
9948 
9991 

9908 
9952 
9996 

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